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(* Title : Series.thy 
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Author : Jacques D. Fleuriot 

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Copyright : 1998 University of Cambridge 

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Converted to Isar and polished by lcp 

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Converted to setsum and polished yet more by TNN 
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Additional contributions by Jeremy Avigad 
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*) 
10751  9 

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header {* Infinite Series *} 
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theory Series 
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Series.thy is based on Limits.thy and not Deriv.thy
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imports Limits 
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begin 
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subsection {* Definition of infinite summability *} 
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definition 
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sums :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a \<Rightarrow> bool" 
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(infixr "sums" 80) 
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where 
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"f sums s \<longleftrightarrow> (\<lambda>n. \<Sum>i<n. f i) > s" 
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definition summable :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> bool" where 
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"summable f \<longleftrightarrow> (\<exists>s. f sums s)" 
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definition 
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suminf :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a" 
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(binder "\<Sum>" 10) 
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where 
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"suminf f = (THE s. f sums s)" 
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subsection {* Infinite summability on topological monoids *} 
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lemma sums_subst[trans]: "f = g \<Longrightarrow> g sums z \<Longrightarrow> f sums z" 
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by simp 
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lemma sums_summable: "f sums l \<Longrightarrow> summable f" 
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by (simp add: sums_def summable_def, blast) 
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lemma summable_iff_convergent: "summable f \<longleftrightarrow> convergent (\<lambda>n. \<Sum>i<n. f i)" 
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by (simp add: summable_def sums_def convergent_def) 
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lemma suminf_eq_lim: "suminf f = lim (\<lambda>n. \<Sum>i<n. f i)" 
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by (simp add: suminf_def sums_def lim_def) 
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lemma sums_zero[simp, intro]: "(\<lambda>n. 0) sums 0" 
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unfolding sums_def by (simp add: tendsto_const) 

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lemma summable_zero[simp, intro]: "summable (\<lambda>n. 0)" 

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by (rule sums_zero [THEN sums_summable]) 

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lemma sums_group: "f sums s \<Longrightarrow> 0 < k \<Longrightarrow> (\<lambda>n. setsum f {n * k ..< n * k + k}) sums s" 

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apply (simp only: sums_def setsum_nat_group tendsto_def eventually_sequentially) 

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apply safe 

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apply (erule_tac x=S in allE) 

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apply safe 

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apply (rule_tac x="N" in exI, safe) 

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apply (drule_tac x="n*k" in spec) 

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apply (erule mp) 

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apply (erule order_trans) 

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apply simp 

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done 

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lemma sums_finite: 
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assumes [simp]: "finite N" and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0" 
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shows "f sums (\<Sum>n\<in>N. f n)" 
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proof  

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{ fix n 

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have "setsum f {..<n + Suc (Max N)} = setsum f N" 

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proof cases 

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assume "N = {}" 

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with f have "f = (\<lambda>x. 0)" by auto 

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then show ?thesis by simp 

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next 

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assume [simp]: "N \<noteq> {}" 

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show ?thesis 

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proof (safe intro!: setsum_mono_zero_right f) 

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fix i assume "i \<in> N" 

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then have "i \<le> Max N" by simp 

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then show "i < n + Suc (Max N)" by simp 

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qed 

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qed } 

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note eq = this 

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show ?thesis unfolding sums_def 

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by (rule LIMSEQ_offset[of _ "Suc (Max N)"]) 

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(simp add: eq atLeast0LessThan tendsto_const del: add_Suc_right) 

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qed 

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lemma summable_finite: "finite N \<Longrightarrow> (\<And>n. n \<notin> N \<Longrightarrow> f n = 0) \<Longrightarrow> summable f" 
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by (rule sums_summable) (rule sums_finite) 

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lemma sums_If_finite_set: "finite A \<Longrightarrow> (\<lambda>r. if r \<in> A then f r else 0) sums (\<Sum>r\<in>A. f r)" 
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using sums_finite[of A "(\<lambda>r. if r \<in> A then f r else 0)"] by simp 
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lemma summable_If_finite_set[simp, intro]: "finite A \<Longrightarrow> summable (\<lambda>r. if r \<in> A then f r else 0)" 
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by (rule sums_summable) (rule sums_If_finite_set) 

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lemma sums_If_finite: "finite {r. P r} \<Longrightarrow> (\<lambda>r. if P r then f r else 0) sums (\<Sum>r  P r. f r)" 
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using sums_If_finite_set[of "{r. P r}"] by simp 
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lemma summable_If_finite[simp, intro]: "finite {r. P r} \<Longrightarrow> summable (\<lambda>r. if P r then f r else 0)" 
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by (rule sums_summable) (rule sums_If_finite) 

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lemma sums_single: "(\<lambda>r. if r = i then f r else 0) sums f i" 
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using sums_If_finite[of "\<lambda>r. r = i"] by simp 
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lemma summable_single[simp, intro]: "summable (\<lambda>r. if r = i then f r else 0)" 
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by (rule sums_summable) (rule sums_single) 

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context 
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fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}" 
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begin 
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lemma summable_sums[intro]: "summable f \<Longrightarrow> f sums (suminf f)" 
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by (simp add: summable_def sums_def suminf_def) 
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(metis convergent_LIMSEQ_iff convergent_def lim_def) 
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lemma summable_LIMSEQ: "summable f \<Longrightarrow> (\<lambda>n. \<Sum>i<n. f i) > suminf f" 
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by (rule summable_sums [unfolded sums_def]) 
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lemma sums_unique: "f sums s \<Longrightarrow> s = suminf f" 
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by (metis limI suminf_eq_lim sums_def) 
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lemma sums_iff: "f sums x \<longleftrightarrow> summable f \<and> (suminf f = x)" 
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by (metis summable_sums sums_summable sums_unique) 
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lemma suminf_finite: 
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assumes N: "finite N" and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0" 
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shows "suminf f = (\<Sum>n\<in>N. f n)" 
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using sums_finite[OF assms, THEN sums_unique] by simp 
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end 
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lemma suminf_zero[simp]: "suminf (\<lambda>n. 0::'a::{t2_space, comm_monoid_add}) = 0" 
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by (rule sums_zero [THEN sums_unique, symmetric]) 
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139 
subsection {* Infinite summability on ordered, topological monoids *} 

140 

141 
lemma sums_le: 

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fixes f g :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology}" 

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shows "\<forall>n. f n \<le> g n \<Longrightarrow> f sums s \<Longrightarrow> g sums t \<Longrightarrow> s \<le> t" 

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by (rule LIMSEQ_le) (auto intro: setsum_mono simp: sums_def) 

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context 
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fixes f :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology}" 
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begin 
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lemma suminf_le: "\<lbrakk>\<forall>n. f n \<le> g n; summable f; summable g\<rbrakk> \<Longrightarrow> suminf f \<le> suminf g" 
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by (auto dest: sums_summable intro: sums_le) 

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lemma setsum_le_suminf: "summable f \<Longrightarrow> \<forall>m\<ge>n. 0 \<le> f m \<Longrightarrow> setsum f {..<n} \<le> suminf f" 

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by (rule sums_le[OF _ sums_If_finite_set summable_sums]) auto 

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lemma suminf_nonneg: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> 0 \<le> suminf f" 

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using setsum_le_suminf[of 0] by simp 

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lemma setsum_less_suminf2: "summable f \<Longrightarrow> \<forall>m\<ge>n. 0 \<le> f m \<Longrightarrow> n \<le> i \<Longrightarrow> 0 < f i \<Longrightarrow> setsum f {..<n} < suminf f" 

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using 

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setsum_le_suminf[of "Suc i"] 

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add_strict_increasing[of "f i" "setsum f {..<n}" "setsum f {..<i}"] 

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setsum_mono2[of "{..<i}" "{..<n}" f] 

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by (auto simp: less_imp_le ac_simps) 

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lemma setsum_less_suminf: "summable f \<Longrightarrow> \<forall>m\<ge>n. 0 < f m \<Longrightarrow> setsum f {..<n} < suminf f" 

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using setsum_less_suminf2[of n n] by (simp add: less_imp_le) 

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lemma suminf_pos2: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> 0 < f i \<Longrightarrow> 0 < suminf f" 

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using setsum_less_suminf2[of 0 i] by simp 

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lemma suminf_pos: "summable f \<Longrightarrow> \<forall>n. 0 < f n \<Longrightarrow> 0 < suminf f" 

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using suminf_pos2[of 0] by (simp add: less_imp_le) 

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lemma suminf_le_const: "summable f \<Longrightarrow> (\<And>n. setsum f {..<n} \<le> x) \<Longrightarrow> suminf f \<le> x" 

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by (metis LIMSEQ_le_const2 summable_LIMSEQ) 

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lemma suminf_eq_zero_iff: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> suminf f = 0 \<longleftrightarrow> (\<forall>n. f n = 0)" 
50999  179 
proof 
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assume "summable f" "suminf f = 0" and pos: "\<forall>n. 0 \<le> f n" 

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then have f: "(\<lambda>n. \<Sum>i<n. f i) > 0" 

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using summable_LIMSEQ[of f] by simp 
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then have "\<And>i. (\<Sum>n\<in>{i}. f n) \<le> 0" 

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proof (rule LIMSEQ_le_const) 

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fix i show "\<exists>N. \<forall>n\<ge>N. (\<Sum>n\<in>{i}. f n) \<le> setsum f {..<n}" 
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using pos by (intro exI[of _ "Suc i"] allI impI setsum_mono2) auto 

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qed 

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with pos show "\<forall>n. f n = 0" 

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by (auto intro!: antisym) 

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qed (metis suminf_zero fun_eq_iff) 
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lemma suminf_pos_iff: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> 0 < suminf f \<longleftrightarrow> (\<exists>i. 0 < f i)" 
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using setsum_le_suminf[of 0] suminf_eq_zero_iff by (simp add: less_le) 

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end 
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lemma summableI_nonneg_bounded: 
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fixes f:: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology, conditionally_complete_linorder}" 

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assumes pos[simp]: "\<And>n. 0 \<le> f n" and le: "\<And>n. (\<Sum>i<n. f i) \<le> x" 

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shows "summable f" 

201 
unfolding summable_def sums_def[abs_def] 

202 
proof (intro exI order_tendstoI) 

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have [simp, intro]: "bdd_above (range (\<lambda>n. \<Sum>i<n. f i))" 

204 
using le by (auto simp: bdd_above_def) 

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{ fix a assume "a < (SUP n. \<Sum>i<n. f i)" 

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then obtain n where "a < (\<Sum>i<n. f i)" 

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by (auto simp add: less_cSUP_iff) 

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then have "\<And>m. n \<le> m \<Longrightarrow> a < (\<Sum>i<m. f i)" 

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by (rule less_le_trans) (auto intro!: setsum_mono2) 

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then show "eventually (\<lambda>n. a < (\<Sum>i<n. f i)) sequentially" 

211 
by (auto simp: eventually_sequentially) } 

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{ fix a assume "(SUP n. \<Sum>i<n. f i) < a" 

213 
moreover have "\<And>n. (\<Sum>i<n. f i) \<le> (SUP n. \<Sum>i<n. f i)" 

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by (auto intro: cSUP_upper) 

215 
ultimately show "eventually (\<lambda>n. (\<Sum>i<n. f i) < a) sequentially" 

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by (auto intro: le_less_trans simp: eventually_sequentially) } 

217 
qed 

218 

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subsection {* Infinite summability on real normed vector spaces *} 

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lemma sums_Suc_iff: 
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fixes f :: "nat \<Rightarrow> 'a::real_normed_vector" 
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shows "(\<lambda>n. f (Suc n)) sums s \<longleftrightarrow> f sums (s + f 0)" 
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proof  
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225 
have "f sums (s + f 0) \<longleftrightarrow> (\<lambda>i. \<Sum>j<Suc i. f j) > s + f 0" 
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226 
by (subst LIMSEQ_Suc_iff) (simp add: sums_def) 
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227 
also have "\<dots> \<longleftrightarrow> (\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) > s + f 0" 
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228 
by (simp add: ac_simps setsum_reindex image_iff lessThan_Suc_eq_insert_0) 
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229 
also have "\<dots> \<longleftrightarrow> (\<lambda>n. f (Suc n)) sums s" 
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230 
proof 
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231 
assume "(\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) > s + f 0" 
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232 
with tendsto_add[OF this tendsto_const, of " f 0"] 
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233 
show "(\<lambda>i. f (Suc i)) sums s" 
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234 
by (simp add: sums_def) 
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qed (auto intro: tendsto_add tendsto_const simp: sums_def) 
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236 
finally show ?thesis .. 
50999  237 
qed 
238 

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239 
context 
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fixes f :: "nat \<Rightarrow> 'a::real_normed_vector" 
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241 
begin 
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242 

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lemma sums_add: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n + g n) sums (a + b)" 
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244 
unfolding sums_def by (simp add: setsum_addf tendsto_add) 
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245 

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246 
lemma summable_add: "summable f \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. f n + g n)" 
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247 
unfolding summable_def by (auto intro: sums_add) 
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248 

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lemma suminf_add: "summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f + suminf g = (\<Sum>n. f n + g n)" 
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by (intro sums_unique sums_add summable_sums) 
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251 

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252 
lemma sums_diff: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n  g n) sums (a  b)" 
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253 
unfolding sums_def by (simp add: setsum_subtractf tendsto_diff) 
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254 

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lemma summable_diff: "summable f \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. f n  g n)" 
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256 
unfolding summable_def by (auto intro: sums_diff) 
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257 

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lemma suminf_diff: "summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f  suminf g = (\<Sum>n. f n  g n)" 
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259 
by (intro sums_unique sums_diff summable_sums) 
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260 

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lemma sums_minus: "f sums a \<Longrightarrow> (\<lambda>n.  f n) sums ( a)" 
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262 
unfolding sums_def by (simp add: setsum_negf tendsto_minus) 
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263 

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lemma summable_minus: "summable f \<Longrightarrow> summable (\<lambda>n.  f n)" 
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265 
unfolding summable_def by (auto intro: sums_minus) 
20692  266 

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lemma suminf_minus: "summable f \<Longrightarrow> (\<Sum>n.  f n) =  (\<Sum>n. f n)" 
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268 
by (intro sums_unique [symmetric] sums_minus summable_sums) 
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269 

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270 
lemma sums_Suc: "(\<lambda> n. f (Suc n)) sums l \<Longrightarrow> f sums (l + f 0)" 
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271 
by (simp add: sums_Suc_iff) 
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272 

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lemma sums_iff_shift: "(\<lambda>i. f (i + n)) sums s \<longleftrightarrow> f sums (s + (\<Sum>i<n. f i))" 
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274 
proof (induct n arbitrary: s) 
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275 
case (Suc n) 
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moreover have "(\<lambda>i. f (Suc i + n)) sums s \<longleftrightarrow> (\<lambda>i. f (i + n)) sums (s + f n)" 
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277 
by (subst sums_Suc_iff) simp 
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278 
ultimately show ?case 
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by (simp add: ac_simps) 
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280 
qed simp 
20692  281 

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lemma summable_iff_shift: "summable (\<lambda>n. f (n + k)) \<longleftrightarrow> summable f" 
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283 
by (metis diff_add_cancel summable_def sums_iff_shift[abs_def]) 
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284 

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285 
lemma sums_split_initial_segment: "f sums s \<Longrightarrow> (\<lambda>i. f (i + n)) sums (s  (\<Sum>i<n. f i))" 
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by (simp add: sums_iff_shift) 
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287 

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lemma summable_ignore_initial_segment: "summable f \<Longrightarrow> summable (\<lambda>n. f(n + k))" 
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by (simp add: summable_iff_shift) 
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290 

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291 
lemma suminf_minus_initial_segment: "summable f \<Longrightarrow> (\<Sum>n. f (n + k)) = (\<Sum>n. f n)  (\<Sum>i<k. f i)" 
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292 
by (rule sums_unique[symmetric]) (auto simp: sums_iff_shift) 
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293 

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294 
lemma suminf_split_initial_segment: "summable f \<Longrightarrow> suminf f = (\<Sum>n. f(n + k)) + (\<Sum>i<k. f i)" 
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295 
by (auto simp add: suminf_minus_initial_segment) 
20692  296 

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297 
lemma suminf_exist_split: 
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298 
fixes r :: real assumes "0 < r" and "summable f" 
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299 
shows "\<exists>N. \<forall>n\<ge>N. norm (\<Sum>i. f (i + n)) < r" 
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300 
proof  
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301 
from LIMSEQ_D[OF summable_LIMSEQ[OF `summable f`] `0 < r`] 
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302 
obtain N :: nat where "\<forall> n \<ge> N. norm (setsum f {..<n}  suminf f) < r" by auto 
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303 
thus ?thesis 
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304 
by (auto simp: norm_minus_commute suminf_minus_initial_segment[OF `summable f`]) 
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305 
qed 
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306 

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307 
lemma summable_LIMSEQ_zero: "summable f \<Longrightarrow> f > 0" 
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308 
apply (drule summable_iff_convergent [THEN iffD1]) 
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309 
apply (drule convergent_Cauchy) 
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310 
apply (simp only: Cauchy_iff LIMSEQ_iff, safe) 
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311 
apply (drule_tac x="r" in spec, safe) 
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312 
apply (rule_tac x="M" in exI, safe) 
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313 
apply (drule_tac x="Suc n" in spec, simp) 
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314 
apply (drule_tac x="n" in spec, simp) 
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315 
done 
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316 

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317 
end 
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318 

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319 
lemma (in bounded_linear) sums: "(\<lambda>n. X n) sums a \<Longrightarrow> (\<lambda>n. f (X n)) sums (f a)" 
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320 
unfolding sums_def by (drule tendsto, simp only: setsum) 
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321 

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322 
lemma (in bounded_linear) summable: "summable (\<lambda>n. X n) \<Longrightarrow> summable (\<lambda>n. f (X n))" 
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323 
unfolding summable_def by (auto intro: sums) 
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324 

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325 
lemma (in bounded_linear) suminf: "summable (\<lambda>n. X n) \<Longrightarrow> f (\<Sum>n. X n) = (\<Sum>n. f (X n))" 
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326 
by (intro sums_unique sums summable_sums) 
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327 

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328 
lemmas sums_of_real = bounded_linear.sums [OF bounded_linear_of_real] 
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329 
lemmas summable_of_real = bounded_linear.summable [OF bounded_linear_of_real] 
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330 
lemmas suminf_of_real = bounded_linear.suminf [OF bounded_linear_of_real] 
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331 

56213  332 
subsection {* Infinite summability on real normed algebras *} 
333 

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334 
context 
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335 
fixes f :: "nat \<Rightarrow> 'a::real_normed_algebra" 
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336 
begin 
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337 

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338 
lemma sums_mult: "f sums a \<Longrightarrow> (\<lambda>n. c * f n) sums (c * a)" 
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339 
by (rule bounded_linear.sums [OF bounded_linear_mult_right]) 
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340 

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341 
lemma summable_mult: "summable f \<Longrightarrow> summable (\<lambda>n. c * f n)" 
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342 
by (rule bounded_linear.summable [OF bounded_linear_mult_right]) 
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343 

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344 
lemma suminf_mult: "summable f \<Longrightarrow> suminf (\<lambda>n. c * f n) = c * suminf f" 
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345 
by (rule bounded_linear.suminf [OF bounded_linear_mult_right, symmetric]) 
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346 

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347 
lemma sums_mult2: "f sums a \<Longrightarrow> (\<lambda>n. f n * c) sums (a * c)" 
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348 
by (rule bounded_linear.sums [OF bounded_linear_mult_left]) 
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349 

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350 
lemma summable_mult2: "summable f \<Longrightarrow> summable (\<lambda>n. f n * c)" 
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351 
by (rule bounded_linear.summable [OF bounded_linear_mult_left]) 
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parents:
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changeset

352 

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353 
lemma suminf_mult2: "summable f \<Longrightarrow> suminf f * c = (\<Sum>n. f n * c)" 
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354 
by (rule bounded_linear.suminf [OF bounded_linear_mult_left]) 
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parents:
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changeset

355 

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diff
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356 
end 
c726ecfb22b6
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357 

56213  358 
subsection {* Infinite summability on real normed fields *} 
359 

56193
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360 
context 
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361 
fixes c :: "'a::real_normed_field" 
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diff
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362 
begin 
c726ecfb22b6
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363 

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364 
lemma sums_divide: "f sums a \<Longrightarrow> (\<lambda>n. f n / c) sums (a / c)" 
c726ecfb22b6
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diff
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365 
by (rule bounded_linear.sums [OF bounded_linear_divide]) 
c726ecfb22b6
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hoelzl
parents:
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diff
changeset

366 

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diff
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367 
lemma summable_divide: "summable f \<Longrightarrow> summable (\<lambda>n. f n / c)" 
c726ecfb22b6
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diff
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368 
by (rule bounded_linear.summable [OF bounded_linear_divide]) 
c726ecfb22b6
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hoelzl
parents:
56178
diff
changeset

369 

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370 
lemma suminf_divide: "summable f \<Longrightarrow> suminf (\<lambda>n. f n / c) = suminf f / c" 
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diff
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371 
by (rule bounded_linear.suminf [OF bounded_linear_divide, symmetric]) 
14416  372 

15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset

373 
text{*Sum of a geometric progression.*} 
14416  374 

56193
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375 
lemma geometric_sums: "norm c < 1 \<Longrightarrow> (\<lambda>n. c^n) sums (1 / (1  c))" 
20692  376 
proof  
56193
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377 
assume less_1: "norm c < 1" 
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diff
changeset

378 
hence neq_1: "c \<noteq> 1" by auto 
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379 
hence neq_0: "c  1 \<noteq> 0" by simp 
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parents:
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changeset

380 
from less_1 have lim_0: "(\<lambda>n. c^n) > 0" 
20692  381 
by (rule LIMSEQ_power_zero) 
56193
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hoelzl
parents:
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diff
changeset

382 
hence "(\<lambda>n. c ^ n / (c  1)  1 / (c  1)) > 0 / (c  1)  1 / (c  1)" 
44568
e6f291cb5810
discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents:
44289
diff
changeset

383 
using neq_0 by (intro tendsto_intros) 
56193
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hoelzl
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384 
hence "(\<lambda>n. (c ^ n  1) / (c  1)) > 1 / (1  c)" 
20692  385 
by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib) 
56193
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386 
thus "(\<lambda>n. c ^ n) sums (1 / (1  c))" 
20692  387 
by (simp add: sums_def geometric_sum neq_1) 
388 
qed 

389 

56193
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parents:
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390 
lemma summable_geometric: "norm c < 1 \<Longrightarrow> summable (\<lambda>n. c^n)" 
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391 
by (rule geometric_sums [THEN sums_summable]) 
14416  392 

56193
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393 
lemma suminf_geometric: "norm c < 1 \<Longrightarrow> suminf (\<lambda>n. c^n) = 1 / (1  c)" 
c726ecfb22b6
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diff
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394 
by (rule sums_unique[symmetric]) (rule geometric_sums) 
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parents:
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diff
changeset

395 

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396 
end 
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

397 

7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

398 
lemma power_half_series: "(\<lambda>n. (1/2::real)^Suc n) sums 1" 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

399 
proof  
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

400 
have 2: "(\<lambda>n. (1/2::real)^n) sums 2" using geometric_sums [of "1/2::real"] 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

401 
by auto 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

402 
have "(\<lambda>n. (1/2::real)^Suc n) = (\<lambda>n. (1 / 2) ^ n / 2)" 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

403 
by simp 
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
41970
diff
changeset

404 
thus ?thesis using sums_divide [OF 2, of 2] 
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

405 
by simp 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

406 
qed 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

407 

56213  408 
subsection {* Infinite summability on Banach spaces *} 
409 

15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset

410 
text{*Cauchytype criterion for convergence of series (c.f. Harrison)*} 
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset

411 

56193
c726ecfb22b6
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parents:
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changeset

412 
lemma summable_Cauchy: 
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changeset

413 
fixes f :: "nat \<Rightarrow> 'a::banach" 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

414 
shows "summable f \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n. norm (setsum f {m..<n}) < e)" 
c726ecfb22b6
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hoelzl
parents:
56178
diff
changeset

415 
apply (simp only: summable_iff_convergent Cauchy_convergent_iff [symmetric] Cauchy_iff, safe) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

416 
apply (drule spec, drule (1) mp) 
c726ecfb22b6
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hoelzl
parents:
56178
diff
changeset

417 
apply (erule exE, rule_tac x="M" in exI, clarify) 
c726ecfb22b6
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hoelzl
parents:
56178
diff
changeset

418 
apply (rule_tac x="m" and y="n" in linorder_le_cases) 
c726ecfb22b6
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hoelzl
parents:
56178
diff
changeset

419 
apply (frule (1) order_trans) 
c726ecfb22b6
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hoelzl
parents:
56178
diff
changeset

420 
apply (drule_tac x="n" in spec, drule (1) mp) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

421 
apply (drule_tac x="m" in spec, drule (1) mp) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

422 
apply (simp_all add: setsum_diff [symmetric]) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

423 
apply (drule spec, drule (1) mp) 
c726ecfb22b6
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hoelzl
parents:
56178
diff
changeset

424 
apply (erule exE, rule_tac x="N" in exI, clarify) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

425 
apply (rule_tac x="m" and y="n" in linorder_le_cases) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

426 
apply (subst norm_minus_commute) 
c726ecfb22b6
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hoelzl
parents:
56178
diff
changeset

427 
apply (simp_all add: setsum_diff [symmetric]) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

428 
done 
14416  429 

56193
c726ecfb22b6
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hoelzl
parents:
56178
diff
changeset

430 
context 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

431 
fixes f :: "nat \<Rightarrow> 'a::banach" 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

432 
begin 
c726ecfb22b6
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hoelzl
parents:
56178
diff
changeset

433 

c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

434 
text{*Absolute convergence imples normal convergence*} 
20689  435 

56194  436 
lemma summable_norm_cancel: "summable (\<lambda>n. norm (f n)) \<Longrightarrow> summable f" 
56193
c726ecfb22b6
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hoelzl
parents:
56178
diff
changeset

437 
apply (simp only: summable_Cauchy, safe) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

438 
apply (drule_tac x="e" in spec, safe) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

439 
apply (rule_tac x="N" in exI, safe) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

440 
apply (drule_tac x="m" in spec, safe) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

441 
apply (rule order_le_less_trans [OF norm_setsum]) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

442 
apply (rule order_le_less_trans [OF abs_ge_self]) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

443 
apply simp 
50999  444 
done 
32707
836ec9d0a0c8
New lemmas involving the real numbers, especially limits and series
paulson
parents:
31336
diff
changeset

445 

56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

446 
lemma summable_norm: "summable (\<lambda>n. norm (f n)) \<Longrightarrow> norm (suminf f) \<le> (\<Sum>n. norm (f n))" 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

447 
by (auto intro: LIMSEQ_le tendsto_norm summable_norm_cancel summable_LIMSEQ norm_setsum) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

448 

c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

449 
text {* Comparison tests *} 
14416  450 

56194  451 
lemma summable_comparison_test: "\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n \<Longrightarrow> summable g \<Longrightarrow> summable f" 
56193
c726ecfb22b6
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hoelzl
parents:
56178
diff
changeset

452 
apply (simp add: summable_Cauchy, safe) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

453 
apply (drule_tac x="e" in spec, safe) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

454 
apply (rule_tac x = "N + Na" in exI, safe) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

455 
apply (rotate_tac 2) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

456 
apply (drule_tac x = m in spec) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

457 
apply (auto, rotate_tac 2, drule_tac x = n in spec) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

458 
apply (rule_tac y = "\<Sum>k=m..<n. norm (f k)" in order_le_less_trans) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

459 
apply (rule norm_setsum) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

460 
apply (rule_tac y = "setsum g {m..<n}" in order_le_less_trans) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

461 
apply (auto intro: setsum_mono simp add: abs_less_iff) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

462 
done 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

463 

56217
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
56213
diff
changeset

464 
(*A better argument order*) 
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
56213
diff
changeset

465 
lemma summable_comparison_test': "summable g \<Longrightarrow> (\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> g n) \<Longrightarrow> summable f" 
56369
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56217
diff
changeset

466 
by (rule summable_comparison_test) auto 
56217
dc429a5b13c4
Some rationalisation of basic lemmas
paulson <lp15@cam.ac.uk>
parents:
56213
diff
changeset

467 

56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

468 
subsection {* The Ratio Test*} 
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset

469 

56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

470 
lemma summable_ratio_test: 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

471 
assumes "c < 1" "\<And>n. n \<ge> N \<Longrightarrow> norm (f (Suc n)) \<le> c * norm (f n)" 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

472 
shows "summable f" 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

473 
proof cases 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

474 
assume "0 < c" 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

475 
show "summable f" 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

476 
proof (rule summable_comparison_test) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

477 
show "\<exists>N'. \<forall>n\<ge>N'. norm (f n) \<le> (norm (f N) / (c ^ N)) * c ^ n" 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

478 
proof (intro exI allI impI) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

479 
fix n assume "N \<le> n" then show "norm (f n) \<le> (norm (f N) / (c ^ N)) * c ^ n" 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

480 
proof (induct rule: inc_induct) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

481 
case (step m) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

482 
moreover have "norm (f (Suc m)) / c ^ Suc m * c ^ n \<le> norm (f m) / c ^ m * c ^ n" 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

483 
using `0 < c` `c < 1` assms(2)[OF `N \<le> m`] by (simp add: field_simps) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

484 
ultimately show ?case by simp 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

485 
qed (insert `0 < c`, simp) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

486 
qed 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

487 
show "summable (\<lambda>n. norm (f N) / c ^ N * c ^ n)" 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

488 
using `0 < c` `c < 1` by (intro summable_mult summable_geometric) simp 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

489 
qed 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

490 
next 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

491 
assume c: "\<not> 0 < c" 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

492 
{ fix n assume "n \<ge> N" 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

493 
then have "norm (f (Suc n)) \<le> c * norm (f n)" 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

494 
by fact 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

495 
also have "\<dots> \<le> 0" 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

496 
using c by (simp add: not_less mult_nonpos_nonneg) 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

497 
finally have "f (Suc n) = 0" 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

498 
by auto } 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

499 
then show "summable f" 
56194  500 
by (intro sums_summable[OF sums_finite, of "{.. Suc N}"]) (auto simp: not_le Suc_less_eq2) 
56178  501 
qed 
502 

56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

503 
end 
14416  504 

56369
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56217
diff
changeset

505 
text{*Relations among convergence and absolute convergence for power series.*} 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56217
diff
changeset

506 

2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56217
diff
changeset

507 
lemma abel_lemma: 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56217
diff
changeset

508 
fixes a :: "nat \<Rightarrow> 'a::real_normed_vector" 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56217
diff
changeset

509 
assumes r: "0 \<le> r" and r0: "r < r0" and M: "\<And>n. norm (a n) * r0^n \<le> M" 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56217
diff
changeset

510 
shows "summable (\<lambda>n. norm (a n) * r^n)" 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56217
diff
changeset

511 
proof (rule summable_comparison_test') 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56217
diff
changeset

512 
show "summable (\<lambda>n. M * (r / r0) ^ n)" 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56217
diff
changeset

513 
using assms 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56217
diff
changeset

514 
by (auto simp add: summable_mult summable_geometric) 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56217
diff
changeset

515 
next 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56217
diff
changeset

516 
fix n 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56217
diff
changeset

517 
show "norm (norm (a n) * r ^ n) \<le> M * (r / r0) ^ n" 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56217
diff
changeset

518 
using r r0 M [of n] 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56217
diff
changeset

519 
apply (auto simp add: abs_mult field_simps power_divide) 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56217
diff
changeset

520 
apply (cases "r=0", simp) 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56217
diff
changeset

521 
apply (cases n, auto) 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56217
diff
changeset

522 
done 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56217
diff
changeset

523 
qed 
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56217
diff
changeset

524 

2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56217
diff
changeset

525 

23084  526 
text{*Summability of geometric series for real algebras*} 
527 

528 
lemma complete_algebra_summable_geometric: 

31017  529 
fixes x :: "'a::{real_normed_algebra_1,banach}" 
23084  530 
shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)" 
531 
proof (rule summable_comparison_test) 

532 
show "\<exists>N. \<forall>n\<ge>N. norm (x ^ n) \<le> norm x ^ n" 

533 
by (simp add: norm_power_ineq) 

534 
show "norm x < 1 \<Longrightarrow> summable (\<lambda>n. norm x ^ n)" 

535 
by (simp add: summable_geometric) 

536 
qed 

537 

23111  538 
subsection {* Cauchy Product Formula *} 
539 

54703  540 
text {* 
541 
Proof based on Analysis WebNotes: Chapter 07, Class 41 

542 
@{url "http://www.math.unl.edu/~webnotes/classes/class41/prp77.htm"} 

543 
*} 

23111  544 

545 
lemma setsum_triangle_reindex: 

546 
fixes n :: nat 

56213  547 
shows "(\<Sum>(i,j)\<in>{(i,j). i+j < n}. f i j) = (\<Sum>k<n. \<Sum>i\<le>k. f i (k  i))" 
23111  548 
proof  
549 
have "(\<Sum>(i, j)\<in>{(i, j). i + j < n}. f i j) = 

56213  550 
(\<Sum>(k, i)\<in>(SIGMA k:{..<n}. {..k}). f i (k  i))" 
23111  551 
proof (rule setsum_reindex_cong) 
56213  552 
show "inj_on (\<lambda>(k,i). (i, k  i)) (SIGMA k:{..<n}. {..k})" 
23111  553 
by (rule inj_on_inverseI [where g="\<lambda>(i,j). (i+j, i)"], auto) 
56213  554 
show "{(i,j). i + j < n} = (\<lambda>(k,i). (i, k  i)) ` (SIGMA k:{..<n}. {..k})" 
23111  555 
by (safe, rule_tac x="(a+b,a)" in image_eqI, auto) 
556 
show "\<And>a. (\<lambda>(k, i). f i (k  i)) a = split f ((\<lambda>(k, i). (i, k  i)) a)" 

557 
by clarify 

558 
qed 

559 
thus ?thesis by (simp add: setsum_Sigma) 

560 
qed 

561 

562 
lemma Cauchy_product_sums: 

563 
fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}" 

564 
assumes a: "summable (\<lambda>k. norm (a k))" 

565 
assumes b: "summable (\<lambda>k. norm (b k))" 

56213  566 
shows "(\<lambda>k. \<Sum>i\<le>k. a i * b (k  i)) sums ((\<Sum>k. a k) * (\<Sum>k. b k))" 
23111  567 
proof  
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

568 
let ?S1 = "\<lambda>n::nat. {..<n} \<times> {..<n}" 
23111  569 
let ?S2 = "\<lambda>n::nat. {(i,j). i + j < n}" 
570 
have S1_mono: "\<And>m n. m \<le> n \<Longrightarrow> ?S1 m \<subseteq> ?S1 n" by auto 

571 
have S2_le_S1: "\<And>n. ?S2 n \<subseteq> ?S1 n" by auto 

572 
have S1_le_S2: "\<And>n. ?S1 (n div 2) \<subseteq> ?S2 n" by auto 

573 
have finite_S1: "\<And>n. finite (?S1 n)" by simp 

574 
with S2_le_S1 have finite_S2: "\<And>n. finite (?S2 n)" by (rule finite_subset) 

575 

576 
let ?g = "\<lambda>(i,j). a i * b j" 

577 
let ?f = "\<lambda>(i,j). norm (a i) * norm (b j)" 

578 
have f_nonneg: "\<And>x. 0 \<le> ?f x" 

579 
by (auto simp add: mult_nonneg_nonneg) 

580 
hence norm_setsum_f: "\<And>A. norm (setsum ?f A) = setsum ?f A" 

581 
unfolding real_norm_def 

582 
by (simp only: abs_of_nonneg setsum_nonneg [rule_format]) 

583 

56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

584 
have "(\<lambda>n. (\<Sum>k<n. a k) * (\<Sum>k<n. b k)) > (\<Sum>k. a k) * (\<Sum>k. b k)" 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

585 
by (intro tendsto_mult summable_LIMSEQ summable_norm_cancel [OF a] summable_norm_cancel [OF b]) 
23111  586 
hence 1: "(\<lambda>n. setsum ?g (?S1 n)) > (\<Sum>k. a k) * (\<Sum>k. b k)" 
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

587 
by (simp only: setsum_product setsum_Sigma [rule_format] finite_lessThan) 
23111  588 

56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

589 
have "(\<lambda>n. (\<Sum>k<n. norm (a k)) * (\<Sum>k<n. norm (b k))) > (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))" 
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

590 
using a b by (intro tendsto_mult summable_LIMSEQ) 
23111  591 
hence "(\<lambda>n. setsum ?f (?S1 n)) > (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))" 
56193
c726ecfb22b6
cleanup Series: sorted according to typeclass hierarchy, use {..<_} instead of {0..<_}
hoelzl
parents:
56178
diff
changeset

592 
by (simp only: setsum_product setsum_Sigma [rule_format] finite_lessThan) 
23111  593 
hence "convergent (\<lambda>n. setsum ?f (?S1 n))" 
594 
by (rule convergentI) 

595 
hence Cauchy: "Cauchy (\<lambda>n. setsum ?f (?S1 n))" 

596 
by (rule convergent_Cauchy) 

36657  597 
have "Zfun (\<lambda>n. setsum ?f (?S1 n  ?S2 n)) sequentially" 
598 
proof (rule ZfunI, simp only: eventually_sequentially norm_setsum_f) 

23111  599 
fix r :: real 
600 
assume r: "0 < r" 

601 
from CauchyD [OF Cauchy r] obtain N 

602 
where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (setsum ?f (?S1 m)  setsum ?f (?S1 n)) < r" .. 

603 
hence "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> norm (setsum ?f (?S1 m  ?S1 n)) < r" 

604 
by (simp only: setsum_diff finite_S1 S1_mono) 

605 
hence N: "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> setsum ?f (?S1 m  ?S1 n) < r" 

606 
by (simp only: norm_setsum_f) 

607 
show "\<exists>N. \<forall>n\<ge>N. setsum ?f (?S1 n  ?S2 n) < r" 

608 
proof (intro exI allI impI) 

609 
fix n assume "2 * N \<le> n" 

610 
hence n: "N \<le> n div 2" by simp 

611 
have "setsum ?f (?S1 n  ?S2 n) \<le> setsum ?f (?S1 n  ?S1 (n div 2))" 

612 
by (intro setsum_mono2 finite_Diff finite_S1 f_nonneg 

613 
Diff_mono subset_refl S1_le_S2) 

614 
also have "\<dots> < r" 

615 
using n div_le_dividend by (rule N) 

616 
finally show "setsum ?f (?S1 n  ?S2 n) < r" . 

617 
qed 

618 
qed 

36657  619 
hence "Zfun (\<lambda>n. setsum ?g (?S1 n  ?S2 n)) sequentially" 
620 
apply (rule Zfun_le [rule_format]) 

23111  621 
apply (simp only: norm_setsum_f) 
622 
apply (rule order_trans [OF norm_setsum setsum_mono]) 

623 
apply (auto simp add: norm_mult_ineq) 

624 
done 

625 
hence 2: "(\<lambda>n. setsum ?g (?S1 n)  setsum ?g (?S2 n)) > 0" 

36660
1cc4ab4b7ff7
make (X > L) an abbreviation for (X > L) sequentially
huffman
parents:
36657
diff
changeset

626 
unfolding tendsto_Zfun_iff diff_0_right 
36657  627 
by (simp only: setsum_diff finite_S1 S2_le_S1) 
23111  628 

629 
with 1 have "(\<lambda>n. setsum ?g (?S2 n)) > (\<Sum>k. a k) * (\<Sum>k. b k)" 

630 
by (rule LIMSEQ_diff_approach_zero2) 

631 
thus ?thesis by (simp only: sums_def setsum_triangle_reindex) 

632 
qed 

633 

634 
lemma Cauchy_product: 

635 
fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}" 

636 
assumes a: "summable (\<lambda>k. norm (a k))" 

637 
assumes b: "summable (\<lambda>k. norm (b k))" 

56213  638 
shows "(\<Sum>k. a k) * (\<Sum>k. b k) = (\<Sum>k. \<Sum>i\<le>k. a i * b (k  i))" 
639 
using a b 

640 
by (rule Cauchy_product_sums [THEN sums_unique]) 

641 

642 
subsection {* Series on @{typ real}s *} 

643 

644 
lemma summable_norm_comparison_test: "\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. norm (f n))" 

645 
by (rule summable_comparison_test) auto 

646 

647 
lemma summable_rabs_comparison_test: "\<lbrakk>\<exists>N. \<forall>n\<ge>N. \<bar>f n\<bar> \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable (\<lambda>n. \<bar>f n :: real\<bar>)" 

648 
by (rule summable_comparison_test) auto 

649 

650 
lemma summable_rabs_cancel: "summable (\<lambda>n. \<bar>f n :: real\<bar>) \<Longrightarrow> summable f" 

651 
by (rule summable_norm_cancel) simp 

652 

653 
lemma summable_rabs: "summable (\<lambda>n. \<bar>f n :: real\<bar>) \<Longrightarrow> \<bar>suminf f\<bar> \<le> (\<Sum>n. \<bar>f n\<bar>)" 

654 
by (fold real_norm_def) (rule summable_norm) 

23111  655 

14416  656 
end 