(* Title : Series.thy Author : Jacques D. Fleuriot Copyright : 1998 University of CambridgeConverted to Isar and polished by lcpConverted to setsum and polished yet more by TNNAdditional contributions by Jeremy Avigad*)section \<open>Infinite Series\<close>theory Seriesimports Limits Inequalitiesbeginsubsection \<open>Definition of infinite summability\<close>definition sums :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a \<Rightarrow> bool" (infixr "sums" 80) where "f sums s \<longleftrightarrow> (\<lambda>n. \<Sum>i<n. f i) \<longlonglongrightarrow> s"definition summable :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> bool" where "summable f \<longleftrightarrow> (\<exists>s. f sums s)"definition suminf :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a" (binder "\<Sum>" 10) where "suminf f = (THE s. f sums s)"text\<open>Variants of the definition\<close>lemma sums_def': "f sums s \<longleftrightarrow> (\<lambda>n. \<Sum>i = 0..n. f i) \<longlonglongrightarrow> s" apply (simp add: sums_def) apply (subst LIMSEQ_Suc_iff [symmetric]) apply (simp only: lessThan_Suc_atMost atLeast0AtMost) donelemma sums_def_le: "f sums s \<longleftrightarrow> (\<lambda>n. \<Sum>i\<le>n. f i) \<longlonglongrightarrow> s" by (simp add: sums_def' atMost_atLeast0)subsection \<open>Infinite summability on topological monoids\<close>lemma sums_subst[trans]: "f = g \<Longrightarrow> g sums z \<Longrightarrow> f sums z" by simplemma sums_cong: "(\<And>n. f n = g n) \<Longrightarrow> f sums c \<longleftrightarrow> g sums c" by (drule ext) simplemma sums_summable: "f sums l \<Longrightarrow> summable f" by (simp add: sums_def summable_def, blast)lemma summable_iff_convergent: "summable f \<longleftrightarrow> convergent (\<lambda>n. \<Sum>i<n. f i)" by (simp add: summable_def sums_def convergent_def)lemma summable_iff_convergent': "summable f \<longleftrightarrow> convergent (\<lambda>n. setsum f {..n})" by (simp_all only: summable_iff_convergent convergent_def lessThan_Suc_atMost [symmetric] LIMSEQ_Suc_iff[of "\<lambda>n. setsum f {..<n}"])lemma suminf_eq_lim: "suminf f = lim (\<lambda>n. \<Sum>i<n. f i)" by (simp add: suminf_def sums_def lim_def)lemma sums_zero[simp, intro]: "(\<lambda>n. 0) sums 0" unfolding sums_def by simplemma summable_zero[simp, intro]: "summable (\<lambda>n. 0)" by (rule sums_zero [THEN sums_summable])lemma sums_group: "f sums s \<Longrightarrow> 0 < k \<Longrightarrow> (\<lambda>n. setsum f {n * k ..< n * k + k}) sums s" apply (simp only: sums_def setsum_nat_group tendsto_def eventually_sequentially) apply safe apply (erule_tac x=S in allE) apply safe apply (rule_tac x="N" in exI, safe) apply (drule_tac x="n*k" in spec) apply (erule mp) apply (erule order_trans) apply simp donelemma suminf_cong: "(\<And>n. f n = g n) \<Longrightarrow> suminf f = suminf g" by (rule arg_cong[of f g], rule ext) simplemma summable_cong: fixes f g :: "nat \<Rightarrow> 'a::real_normed_vector" assumes "eventually (\<lambda>x. f x = g x) sequentially" shows "summable f = summable g"proof - from assms obtain N where N: "\<forall>n\<ge>N. f n = g n" by (auto simp: eventually_at_top_linorder) define C where "C = (\<Sum>k<N. f k - g k)" from eventually_ge_at_top[of N] have "eventually (\<lambda>n. setsum f {..<n} = C + setsum g {..<n}) sequentially" proof eventually_elim case (elim n) then have "{..<n} = {..<N} \<union> {N..<n}" by auto also have "setsum f ... = setsum f {..<N} + setsum f {N..<n}" by (intro setsum.union_disjoint) auto also from N have "setsum f {N..<n} = setsum g {N..<n}" by (intro setsum.cong) simp_all also have "setsum f {..<N} + setsum g {N..<n} = C + (setsum g {..<N} + setsum g {N..<n})" unfolding C_def by (simp add: algebra_simps setsum_subtractf) also have "setsum g {..<N} + setsum g {N..<n} = setsum g ({..<N} \<union> {N..<n})" by (intro setsum.union_disjoint [symmetric]) auto also from elim have "{..<N} \<union> {N..<n} = {..<n}" by auto finally show "setsum f {..<n} = C + setsum g {..<n}" . qed from convergent_cong[OF this] show ?thesis by (simp add: summable_iff_convergent convergent_add_const_iff)qedlemma sums_finite: assumes [simp]: "finite N" and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0" shows "f sums (\<Sum>n\<in>N. f n)"proof - have eq: "setsum f {..<n + Suc (Max N)} = setsum f N" for n proof (cases "N = {}") case True with f have "f = (\<lambda>x. 0)" by auto then show ?thesis by simp next case [simp]: False show ?thesis proof (safe intro!: setsum.mono_neutral_right f) fix i assume "i \<in> N" then have "i \<le> Max N" by simp then show "i < n + Suc (Max N)" by simp qed qed show ?thesis unfolding sums_def by (rule LIMSEQ_offset[of _ "Suc (Max N)"]) (simp add: eq atLeast0LessThan del: add_Suc_right)qedcorollary sums_0: "(\<And>n. f n = 0) \<Longrightarrow> (f sums 0)" by (metis (no_types) finite.emptyI setsum.empty sums_finite)lemma summable_finite: "finite N \<Longrightarrow> (\<And>n. n \<notin> N \<Longrightarrow> f n = 0) \<Longrightarrow> summable f" by (rule sums_summable) (rule sums_finite)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)" using sums_finite[of A "(\<lambda>r. if r \<in> A then f r else 0)"] by simplemma summable_If_finite_set[simp, intro]: "finite A \<Longrightarrow> summable (\<lambda>r. if r \<in> A then f r else 0)" by (rule sums_summable) (rule sums_If_finite_set)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)" using sums_If_finite_set[of "{r. P r}"] by simplemma summable_If_finite[simp, intro]: "finite {r. P r} \<Longrightarrow> summable (\<lambda>r. if P r then f r else 0)" by (rule sums_summable) (rule sums_If_finite)lemma sums_single: "(\<lambda>r. if r = i then f r else 0) sums f i" using sums_If_finite[of "\<lambda>r. r = i"] by simplemma summable_single[simp, intro]: "summable (\<lambda>r. if r = i then f r else 0)" by (rule sums_summable) (rule sums_single)context fixes f :: "nat \<Rightarrow> 'a::{t2_space,comm_monoid_add}"beginlemma summable_sums[intro]: "summable f \<Longrightarrow> f sums (suminf f)" by (simp add: summable_def sums_def suminf_def) (metis convergent_LIMSEQ_iff convergent_def lim_def)lemma summable_LIMSEQ: "summable f \<Longrightarrow> (\<lambda>n. \<Sum>i<n. f i) \<longlonglongrightarrow> suminf f" by (rule summable_sums [unfolded sums_def])lemma sums_unique: "f sums s \<Longrightarrow> s = suminf f" by (metis limI suminf_eq_lim sums_def)lemma sums_iff: "f sums x \<longleftrightarrow> summable f \<and> suminf f = x" by (metis summable_sums sums_summable sums_unique)lemma summable_sums_iff: "summable f \<longleftrightarrow> f sums suminf f" by (auto simp: sums_iff summable_sums)lemma sums_unique2: "f sums a \<Longrightarrow> f sums b \<Longrightarrow> a = b" for a b :: 'a by (simp add: sums_iff)lemma suminf_finite: assumes N: "finite N" and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0" shows "suminf f = (\<Sum>n\<in>N. f n)" using sums_finite[OF assms, THEN sums_unique] by simpendlemma suminf_zero[simp]: "suminf (\<lambda>n. 0::'a::{t2_space, comm_monoid_add}) = 0" by (rule sums_zero [THEN sums_unique, symmetric])subsection \<open>Infinite summability on ordered, topological monoids\<close>lemma sums_le: "\<forall>n. f n \<le> g n \<Longrightarrow> f sums s \<Longrightarrow> g sums t \<Longrightarrow> s \<le> t" for f g :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add,linorder_topology}" by (rule LIMSEQ_le) (auto intro: setsum_mono simp: sums_def)context fixes f :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add,linorder_topology}"beginlemma suminf_le: "\<forall>n. f n \<le> g n \<Longrightarrow> summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f \<le> suminf g" by (auto dest: sums_summable intro: sums_le)lemma setsum_le_suminf: "summable f \<Longrightarrow> \<forall>m\<ge>n. 0 \<le> f m \<Longrightarrow> setsum f {..<n} \<le> suminf f" by (rule sums_le[OF _ sums_If_finite_set summable_sums]) autolemma suminf_nonneg: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> 0 \<le> suminf f" using setsum_le_suminf[of 0] by simplemma suminf_le_const: "summable f \<Longrightarrow> (\<And>n. setsum f {..<n} \<le> x) \<Longrightarrow> suminf f \<le> x" by (metis LIMSEQ_le_const2 summable_LIMSEQ)lemma suminf_eq_zero_iff: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> suminf f = 0 \<longleftrightarrow> (\<forall>n. f n = 0)"proof assume "summable f" "suminf f = 0" and pos: "\<forall>n. 0 \<le> f n" then have f: "(\<lambda>n. \<Sum>i<n. f i) \<longlonglongrightarrow> 0" using summable_LIMSEQ[of f] by simp then have "\<And>i. (\<Sum>n\<in>{i}. f n) \<le> 0" proof (rule LIMSEQ_le_const) show "\<exists>N. \<forall>n\<ge>N. (\<Sum>n\<in>{i}. f n) \<le> setsum f {..<n}" for i using pos by (intro exI[of _ "Suc i"] allI impI setsum_mono2) auto qed with pos show "\<forall>n. f n = 0" by (auto intro!: antisym)qed (metis suminf_zero fun_eq_iff)lemma suminf_pos_iff: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> 0 < suminf f \<longleftrightarrow> (\<exists>i. 0 < f i)" using setsum_le_suminf[of 0] suminf_eq_zero_iff by (simp add: less_le)lemma suminf_pos2: assumes "summable f" "\<forall>n. 0 \<le> f n" "0 < f i" shows "0 < suminf f"proof - have "0 < (\<Sum>n<Suc i. f n)" using assms by (intro setsum_pos2[where i=i]) auto also have "\<dots> \<le> suminf f" using assms by (intro setsum_le_suminf) auto finally show ?thesis .qedlemma suminf_pos: "summable f \<Longrightarrow> \<forall>n. 0 < f n \<Longrightarrow> 0 < suminf f" by (intro suminf_pos2[where i=0]) (auto intro: less_imp_le)endcontext fixes f :: "nat \<Rightarrow> 'a::{ordered_cancel_comm_monoid_add,linorder_topology}"beginlemma 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" using setsum_le_suminf[of f "Suc i"] and add_strict_increasing[of "f i" "setsum f {..<n}" "setsum f {..<i}"] and setsum_mono2[of "{..<i}" "{..<n}" f] by (auto simp: less_imp_le ac_simps)lemma setsum_less_suminf: "summable f \<Longrightarrow> \<forall>m\<ge>n. 0 < f m \<Longrightarrow> setsum f {..<n} < suminf f" using setsum_less_suminf2[of n n] by (simp add: less_imp_le)endlemma summableI_nonneg_bounded: fixes f :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add,linorder_topology,conditionally_complete_linorder}" assumes pos[simp]: "\<And>n. 0 \<le> f n" and le: "\<And>n. (\<Sum>i<n. f i) \<le> x" shows "summable f" unfolding summable_def sums_def [abs_def]proof (rule exI LIMSEQ_incseq_SUP)+ show "bdd_above (range (\<lambda>n. setsum f {..<n}))" using le by (auto simp: bdd_above_def) show "incseq (\<lambda>n. setsum f {..<n})" by (auto simp: mono_def intro!: setsum_mono2)qedlemma summableI[intro, simp]: "summable f" for f :: "nat \<Rightarrow> 'a::{canonically_ordered_monoid_add,linorder_topology,complete_linorder}" by (intro summableI_nonneg_bounded[where x=top] zero_le top_greatest)subsection \<open>Infinite summability on topological monoids\<close>context fixes f g :: "nat \<Rightarrow> 'a::{t2_space,topological_comm_monoid_add}"beginlemma sums_Suc: assumes "(\<lambda>n. f (Suc n)) sums l" shows "f sums (l + f 0)"proof - have "(\<lambda>n. (\<Sum>i<n. f (Suc i)) + f 0) \<longlonglongrightarrow> l + f 0" using assms by (auto intro!: tendsto_add simp: sums_def) moreover have "(\<Sum>i<n. f (Suc i)) + f 0 = (\<Sum>i<Suc n. f i)" for n unfolding lessThan_Suc_eq_insert_0 by (simp add: ac_simps setsum_atLeast1_atMost_eq image_Suc_lessThan) ultimately show ?thesis by (auto simp: sums_def simp del: setsum_lessThan_Suc intro: LIMSEQ_Suc_iff[THEN iffD1])qedlemma sums_add: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n + g n) sums (a + b)" unfolding sums_def by (simp add: setsum.distrib tendsto_add)lemma summable_add: "summable f \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. f n + g n)" unfolding summable_def by (auto intro: sums_add)lemma suminf_add: "summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f + suminf g = (\<Sum>n. f n + g n)" by (intro sums_unique sums_add summable_sums)endcontext fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::{t2_space,topological_comm_monoid_add}" and I :: "'i set"beginlemma sums_setsum: "(\<And>i. i \<in> I \<Longrightarrow> (f i) sums (x i)) \<Longrightarrow> (\<lambda>n. \<Sum>i\<in>I. f i n) sums (\<Sum>i\<in>I. x i)" by (induct I rule: infinite_finite_induct) (auto intro!: sums_add)lemma suminf_setsum: "(\<And>i. i \<in> I \<Longrightarrow> summable (f i)) \<Longrightarrow> (\<Sum>n. \<Sum>i\<in>I. f i n) = (\<Sum>i\<in>I. \<Sum>n. f i n)" using sums_unique[OF sums_setsum, OF summable_sums] by simplemma summable_setsum: "(\<And>i. i \<in> I \<Longrightarrow> summable (f i)) \<Longrightarrow> summable (\<lambda>n. \<Sum>i\<in>I. f i n)" using sums_summable[OF sums_setsum[OF summable_sums]] .endsubsection \<open>Infinite summability on real normed vector spaces\<close>context fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"beginlemma sums_Suc_iff: "(\<lambda>n. f (Suc n)) sums s \<longleftrightarrow> f sums (s + f 0)"proof - have "f sums (s + f 0) \<longleftrightarrow> (\<lambda>i. \<Sum>j<Suc i. f j) \<longlonglongrightarrow> s + f 0" by (subst LIMSEQ_Suc_iff) (simp add: sums_def) also have "\<dots> \<longleftrightarrow> (\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) \<longlonglongrightarrow> s + f 0" by (simp add: ac_simps lessThan_Suc_eq_insert_0 image_Suc_lessThan setsum_atLeast1_atMost_eq) also have "\<dots> \<longleftrightarrow> (\<lambda>n. f (Suc n)) sums s" proof assume "(\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) \<longlonglongrightarrow> s + f 0" with tendsto_add[OF this tendsto_const, of "- f 0"] show "(\<lambda>i. f (Suc i)) sums s" by (simp add: sums_def) qed (auto intro: tendsto_add simp: sums_def) finally show ?thesis ..qedlemma summable_Suc_iff: "summable (\<lambda>n. f (Suc n)) = summable f"proof assume "summable f" then have "f sums suminf f" by (rule summable_sums) then have "(\<lambda>n. f (Suc n)) sums (suminf f - f 0)" by (simp add: sums_Suc_iff) then show "summable (\<lambda>n. f (Suc n))" unfolding summable_def by blastqed (auto simp: sums_Suc_iff summable_def)lemma sums_Suc_imp: "f 0 = 0 \<Longrightarrow> (\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s" using sums_Suc_iff by simpendcontext (* Separate contexts are necessary to allow general use of the results above, here. *) fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"beginlemma sums_diff: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n - g n) sums (a - b)" unfolding sums_def by (simp add: setsum_subtractf tendsto_diff)lemma summable_diff: "summable f \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. f n - g n)" unfolding summable_def by (auto intro: sums_diff)lemma suminf_diff: "summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f - suminf g = (\<Sum>n. f n - g n)" by (intro sums_unique sums_diff summable_sums)lemma sums_minus: "f sums a \<Longrightarrow> (\<lambda>n. - f n) sums (- a)" unfolding sums_def by (simp add: setsum_negf tendsto_minus)lemma summable_minus: "summable f \<Longrightarrow> summable (\<lambda>n. - f n)" unfolding summable_def by (auto intro: sums_minus)lemma suminf_minus: "summable f \<Longrightarrow> (\<Sum>n. - f n) = - (\<Sum>n. f n)" by (intro sums_unique [symmetric] sums_minus summable_sums)lemma sums_iff_shift: "(\<lambda>i. f (i + n)) sums s \<longleftrightarrow> f sums (s + (\<Sum>i<n. f i))"proof (induct n arbitrary: s) case 0 then show ?case by simpnext case (Suc n) then have "(\<lambda>i. f (Suc i + n)) sums s \<longleftrightarrow> (\<lambda>i. f (i + n)) sums (s + f n)" by (subst sums_Suc_iff) simp with Suc show ?case by (simp add: ac_simps)qedcorollary sums_iff_shift': "(\<lambda>i. f (i + n)) sums (s - (\<Sum>i<n. f i)) \<longleftrightarrow> f sums s" by (simp add: sums_iff_shift)lemma sums_zero_iff_shift: assumes "\<And>i. i < n \<Longrightarrow> f i = 0" shows "(\<lambda>i. f (i+n)) sums s \<longleftrightarrow> (\<lambda>i. f i) sums s" by (simp add: assms sums_iff_shift)lemma summable_iff_shift: "summable (\<lambda>n. f (n + k)) \<longleftrightarrow> summable f" by (metis diff_add_cancel summable_def sums_iff_shift [abs_def])lemma sums_split_initial_segment: "f sums s \<Longrightarrow> (\<lambda>i. f (i + n)) sums (s - (\<Sum>i<n. f i))" by (simp add: sums_iff_shift)lemma summable_ignore_initial_segment: "summable f \<Longrightarrow> summable (\<lambda>n. f(n + k))" by (simp add: summable_iff_shift)lemma suminf_minus_initial_segment: "summable f \<Longrightarrow> (\<Sum>n. f (n + k)) = (\<Sum>n. f n) - (\<Sum>i<k. f i)" by (rule sums_unique[symmetric]) (auto simp: sums_iff_shift)lemma suminf_split_initial_segment: "summable f \<Longrightarrow> suminf f = (\<Sum>n. f(n + k)) + (\<Sum>i<k. f i)" by (auto simp add: suminf_minus_initial_segment)lemma suminf_split_head: "summable f \<Longrightarrow> (\<Sum>n. f (Suc n)) = suminf f - f 0" using suminf_split_initial_segment[of 1] by simplemma suminf_exist_split: fixes r :: real assumes "0 < r" and "summable f" shows "\<exists>N. \<forall>n\<ge>N. norm (\<Sum>i. f (i + n)) < r"proof - from LIMSEQ_D[OF summable_LIMSEQ[OF \<open>summable f\<close>] \<open>0 < r\<close>] obtain N :: nat where "\<forall> n \<ge> N. norm (setsum f {..<n} - suminf f) < r" by auto then show ?thesis by (auto simp: norm_minus_commute suminf_minus_initial_segment[OF \<open>summable f\<close>])qedlemma summable_LIMSEQ_zero: "summable f \<Longrightarrow> f \<longlonglongrightarrow> 0" apply (drule summable_iff_convergent [THEN iffD1]) apply (drule convergent_Cauchy) apply (simp only: Cauchy_iff LIMSEQ_iff) apply safe apply (drule_tac x="r" in spec) apply safe apply (rule_tac x="M" in exI) apply safe apply (drule_tac x="Suc n" in spec) apply simp apply (drule_tac x="n" in spec) apply simp donelemma summable_imp_convergent: "summable f \<Longrightarrow> convergent f" by (force dest!: summable_LIMSEQ_zero simp: convergent_def)lemma summable_imp_Bseq: "summable f \<Longrightarrow> Bseq f" by (simp add: convergent_imp_Bseq summable_imp_convergent)endlemma summable_minus_iff: "summable (\<lambda>n. - f n) \<longleftrightarrow> summable f" for f :: "nat \<Rightarrow> 'a::real_normed_vector" by (auto dest: summable_minus) (* used two ways, hence must be outside the context above *)lemma (in bounded_linear) sums: "(\<lambda>n. X n) sums a \<Longrightarrow> (\<lambda>n. f (X n)) sums (f a)" unfolding sums_def by (drule tendsto) (simp only: setsum)lemma (in bounded_linear) summable: "summable (\<lambda>n. X n) \<Longrightarrow> summable (\<lambda>n. f (X n))" unfolding summable_def by (auto intro: sums)lemma (in bounded_linear) suminf: "summable (\<lambda>n. X n) \<Longrightarrow> f (\<Sum>n. X n) = (\<Sum>n. f (X n))" by (intro sums_unique sums summable_sums)lemmas sums_of_real = bounded_linear.sums [OF bounded_linear_of_real]lemmas summable_of_real = bounded_linear.summable [OF bounded_linear_of_real]lemmas suminf_of_real = bounded_linear.suminf [OF bounded_linear_of_real]lemmas sums_scaleR_left = bounded_linear.sums[OF bounded_linear_scaleR_left]lemmas summable_scaleR_left = bounded_linear.summable[OF bounded_linear_scaleR_left]lemmas suminf_scaleR_left = bounded_linear.suminf[OF bounded_linear_scaleR_left]lemmas sums_scaleR_right = bounded_linear.sums[OF bounded_linear_scaleR_right]lemmas summable_scaleR_right = bounded_linear.summable[OF bounded_linear_scaleR_right]lemmas suminf_scaleR_right = bounded_linear.suminf[OF bounded_linear_scaleR_right]lemma summable_const_iff: "summable (\<lambda>_. c) \<longleftrightarrow> c = 0" for c :: "'a::real_normed_vector"proof - have "\<not> summable (\<lambda>_. c)" if "c \<noteq> 0" proof - from that have "filterlim (\<lambda>n. of_nat n * norm c) at_top sequentially" by (subst mult.commute) (auto intro!: filterlim_tendsto_pos_mult_at_top filterlim_real_sequentially) then have "\<not> convergent (\<lambda>n. norm (\<Sum>k<n. c))" by (intro filterlim_at_infinity_imp_not_convergent filterlim_at_top_imp_at_infinity) (simp_all add: setsum_constant_scaleR) then show ?thesis unfolding summable_iff_convergent using convergent_norm by blast qed then show ?thesis by autoqedsubsection \<open>Infinite summability on real normed algebras\<close>context fixes f :: "nat \<Rightarrow> 'a::real_normed_algebra"beginlemma sums_mult: "f sums a \<Longrightarrow> (\<lambda>n. c * f n) sums (c * a)" by (rule bounded_linear.sums [OF bounded_linear_mult_right])lemma summable_mult: "summable f \<Longrightarrow> summable (\<lambda>n. c * f n)" by (rule bounded_linear.summable [OF bounded_linear_mult_right])lemma suminf_mult: "summable f \<Longrightarrow> suminf (\<lambda>n. c * f n) = c * suminf f" by (rule bounded_linear.suminf [OF bounded_linear_mult_right, symmetric])lemma sums_mult2: "f sums a \<Longrightarrow> (\<lambda>n. f n * c) sums (a * c)" by (rule bounded_linear.sums [OF bounded_linear_mult_left])lemma summable_mult2: "summable f \<Longrightarrow> summable (\<lambda>n. f n * c)" by (rule bounded_linear.summable [OF bounded_linear_mult_left])lemma suminf_mult2: "summable f \<Longrightarrow> suminf f * c = (\<Sum>n. f n * c)" by (rule bounded_linear.suminf [OF bounded_linear_mult_left])endlemma sums_mult_iff: fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra,field}" assumes "c \<noteq> 0" shows "(\<lambda>n. c * f n) sums (c * d) \<longleftrightarrow> f sums d" using sums_mult[of f d c] sums_mult[of "\<lambda>n. c * f n" "c * d" "inverse c"] by (force simp: field_simps assms)lemma sums_mult2_iff: fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra,field}" assumes "c \<noteq> 0" shows "(\<lambda>n. f n * c) sums (d * c) \<longleftrightarrow> f sums d" using sums_mult_iff[OF assms, of f d] by (simp add: mult.commute)lemma sums_of_real_iff: "(\<lambda>n. of_real (f n) :: 'a::real_normed_div_algebra) sums of_real c \<longleftrightarrow> f sums c" by (simp add: sums_def of_real_setsum[symmetric] tendsto_of_real_iff del: of_real_setsum)subsection \<open>Infinite summability on real normed fields\<close>context fixes c :: "'a::real_normed_field"beginlemma sums_divide: "f sums a \<Longrightarrow> (\<lambda>n. f n / c) sums (a / c)" by (rule bounded_linear.sums [OF bounded_linear_divide])lemma summable_divide: "summable f \<Longrightarrow> summable (\<lambda>n. f n / c)" by (rule bounded_linear.summable [OF bounded_linear_divide])lemma suminf_divide: "summable f \<Longrightarrow> suminf (\<lambda>n. f n / c) = suminf f / c" by (rule bounded_linear.suminf [OF bounded_linear_divide, symmetric])lemma sums_mult_D: "(\<lambda>n. c * f n) sums a \<Longrightarrow> c \<noteq> 0 \<Longrightarrow> f sums (a/c)" using sums_mult_iff by fastforcelemma summable_mult_D: "summable (\<lambda>n. c * f n) \<Longrightarrow> c \<noteq> 0 \<Longrightarrow> summable f" by (auto dest: summable_divide)text \<open>Sum of a geometric progression.\<close>lemma geometric_sums: assumes less_1: "norm c < 1" shows "(\<lambda>n. c^n) sums (1 / (1 - c))"proof - from less_1 have neq_1: "c \<noteq> 1" by auto then have neq_0: "c - 1 \<noteq> 0" by simp from less_1 have lim_0: "(\<lambda>n. c^n) \<longlonglongrightarrow> 0" by (rule LIMSEQ_power_zero) then have "(\<lambda>n. c ^ n / (c - 1) - 1 / (c - 1)) \<longlonglongrightarrow> 0 / (c - 1) - 1 / (c - 1)" using neq_0 by (intro tendsto_intros) then have "(\<lambda>n. (c ^ n - 1) / (c - 1)) \<longlonglongrightarrow> 1 / (1 - c)" by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib) then show "(\<lambda>n. c ^ n) sums (1 / (1 - c))" by (simp add: sums_def geometric_sum neq_1)qedlemma summable_geometric: "norm c < 1 \<Longrightarrow> summable (\<lambda>n. c^n)" by (rule geometric_sums [THEN sums_summable])lemma suminf_geometric: "norm c < 1 \<Longrightarrow> suminf (\<lambda>n. c^n) = 1 / (1 - c)" by (rule sums_unique[symmetric]) (rule geometric_sums)lemma summable_geometric_iff: "summable (\<lambda>n. c ^ n) \<longleftrightarrow> norm c < 1"proof assume "summable (\<lambda>n. c ^ n :: 'a :: real_normed_field)" then have "(\<lambda>n. norm c ^ n) \<longlonglongrightarrow> 0" by (simp add: norm_power [symmetric] tendsto_norm_zero_iff summable_LIMSEQ_zero) from order_tendstoD(2)[OF this zero_less_one] obtain n where "norm c ^ n < 1" by (auto simp: eventually_at_top_linorder) then show "norm c < 1" using one_le_power[of "norm c" n] by (cases "norm c \<ge> 1") (linarith, simp)qed (rule summable_geometric)endlemma power_half_series: "(\<lambda>n. (1/2::real)^Suc n) sums 1"proof - have 2: "(\<lambda>n. (1/2::real)^n) sums 2" using geometric_sums [of "1/2::real"] by auto have "(\<lambda>n. (1/2::real)^Suc n) = (\<lambda>n. (1 / 2) ^ n / 2)" by (simp add: mult.commute) then show ?thesis using sums_divide [OF 2, of 2] by simpqedsubsection \<open>Telescoping\<close>lemma telescope_sums: fixes c :: "'a::real_normed_vector" assumes "f \<longlonglongrightarrow> c" shows "(\<lambda>n. f (Suc n) - f n) sums (c - f 0)" unfolding sums_defproof (subst LIMSEQ_Suc_iff [symmetric]) have "(\<lambda>n. \<Sum>k<Suc n. f (Suc k) - f k) = (\<lambda>n. f (Suc n) - f 0)" by (simp add: lessThan_Suc_atMost atLeast0AtMost [symmetric] setsum_Suc_diff) also have "\<dots> \<longlonglongrightarrow> c - f 0" by (intro tendsto_diff LIMSEQ_Suc[OF assms] tendsto_const) finally show "(\<lambda>n. \<Sum>n<Suc n. f (Suc n) - f n) \<longlonglongrightarrow> c - f 0" .qedlemma telescope_sums': fixes c :: "'a::real_normed_vector" assumes "f \<longlonglongrightarrow> c" shows "(\<lambda>n. f n - f (Suc n)) sums (f 0 - c)" using sums_minus[OF telescope_sums[OF assms]] by (simp add: algebra_simps)lemma telescope_summable: fixes c :: "'a::real_normed_vector" assumes "f \<longlonglongrightarrow> c" shows "summable (\<lambda>n. f (Suc n) - f n)" using telescope_sums[OF assms] by (simp add: sums_iff)lemma telescope_summable': fixes c :: "'a::real_normed_vector" assumes "f \<longlonglongrightarrow> c" shows "summable (\<lambda>n. f n - f (Suc n))" using summable_minus[OF telescope_summable[OF assms]] by (simp add: algebra_simps)subsection \<open>Infinite summability on Banach spaces\<close>text \<open>Cauchy-type criterion for convergence of series (c.f. Harrison).\<close>lemma summable_Cauchy: "summable f \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n. norm (setsum f {m..<n}) < e)" for f :: "nat \<Rightarrow> 'a::banach" apply (simp only: summable_iff_convergent Cauchy_convergent_iff [symmetric] Cauchy_iff) apply safe apply (drule spec) apply (drule (1) mp) apply (erule exE) apply (rule_tac x="M" in exI) apply clarify apply (rule_tac x="m" and y="n" in linorder_le_cases) apply (frule (1) order_trans) apply (drule_tac x="n" in spec) apply (drule (1) mp) apply (drule_tac x="m" in spec) apply (drule (1) mp) apply (simp_all add: setsum_diff [symmetric]) apply (drule spec) apply (drule (1) mp) apply (erule exE) apply (rule_tac x="N" in exI) apply clarify apply (rule_tac x="m" and y="n" in linorder_le_cases) apply (subst norm_minus_commute) apply (simp_all add: setsum_diff [symmetric]) donecontext fixes f :: "nat \<Rightarrow> 'a::banach"begintext \<open>Absolute convergence imples normal convergence.\<close>lemma summable_norm_cancel: "summable (\<lambda>n. norm (f n)) \<Longrightarrow> summable f" apply (simp only: summable_Cauchy) apply safe apply (drule_tac x="e" in spec) apply safe apply (rule_tac x="N" in exI) apply safe apply (drule_tac x="m" in spec) apply safe apply (rule order_le_less_trans [OF norm_setsum]) apply (rule order_le_less_trans [OF abs_ge_self]) apply simp donelemma summable_norm: "summable (\<lambda>n. norm (f n)) \<Longrightarrow> norm (suminf f) \<le> (\<Sum>n. norm (f n))" by (auto intro: LIMSEQ_le tendsto_norm summable_norm_cancel summable_LIMSEQ norm_setsum)text \<open>Comparison tests.\<close>lemma summable_comparison_test: "\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n \<Longrightarrow> summable g \<Longrightarrow> summable f" apply (simp add: summable_Cauchy) apply safe apply (drule_tac x="e" in spec) apply safe apply (rule_tac x = "N + Na" in exI) apply safe apply (rotate_tac 2) apply (drule_tac x = m in spec) apply auto apply (rotate_tac 2) apply (drule_tac x = n in spec) apply (rule_tac y = "\<Sum>k=m..<n. norm (f k)" in order_le_less_trans) apply (rule norm_setsum) apply (rule_tac y = "setsum g {m..<n}" in order_le_less_trans) apply (auto intro: setsum_mono simp add: abs_less_iff) donelemma summable_comparison_test_ev: "eventually (\<lambda>n. norm (f n) \<le> g n) sequentially \<Longrightarrow> summable g \<Longrightarrow> summable f" by (rule summable_comparison_test) (auto simp: eventually_at_top_linorder)text \<open>A better argument order.\<close>lemma summable_comparison_test': "summable g \<Longrightarrow> (\<And>n. n \<ge> N \<Longrightarrow> norm (f n) \<le> g n) \<Longrightarrow> summable f" by (rule summable_comparison_test) autosubsection \<open>The Ratio Test\<close>lemma summable_ratio_test: assumes "c < 1" "\<And>n. n \<ge> N \<Longrightarrow> norm (f (Suc n)) \<le> c * norm (f n)" shows "summable f"proof (cases "0 < c") case True show "summable f" proof (rule summable_comparison_test) show "\<exists>N'. \<forall>n\<ge>N'. norm (f n) \<le> (norm (f N) / (c ^ N)) * c ^ n" proof (intro exI allI impI) fix n assume "N \<le> n" then show "norm (f n) \<le> (norm (f N) / (c ^ N)) * c ^ n" proof (induct rule: inc_induct) case base with True show ?case by simp next case (step m) have "norm (f (Suc m)) / c ^ Suc m * c ^ n \<le> norm (f m) / c ^ m * c ^ n" using \<open>0 < c\<close> \<open>c < 1\<close> assms(2)[OF \<open>N \<le> m\<close>] by (simp add: field_simps) with step show ?case by simp qed qed show "summable (\<lambda>n. norm (f N) / c ^ N * c ^ n)" using \<open>0 < c\<close> \<open>c < 1\<close> by (intro summable_mult summable_geometric) simp qednext case False have "f (Suc n) = 0" if "n \<ge> N" for n proof - from that have "norm (f (Suc n)) \<le> c * norm (f n)" by (rule assms(2)) also have "\<dots> \<le> 0" using False by (simp add: not_less mult_nonpos_nonneg) finally show ?thesis by auto qed then show "summable f" by (intro sums_summable[OF sums_finite, of "{.. Suc N}"]) (auto simp: not_le Suc_less_eq2)qedendtext \<open>Relations among convergence and absolute convergence for power series.\<close>lemma Abel_lemma: fixes a :: "nat \<Rightarrow> 'a::real_normed_vector" assumes r: "0 \<le> r" and r0: "r < r0" and M: "\<And>n. norm (a n) * r0^n \<le> M" shows "summable (\<lambda>n. norm (a n) * r^n)"proof (rule summable_comparison_test') show "summable (\<lambda>n. M * (r / r0) ^ n)" using assms by (auto simp add: summable_mult summable_geometric) show "norm (norm (a n) * r ^ n) \<le> M * (r / r0) ^ n" for n using r r0 M [of n] apply (auto simp add: abs_mult field_simps) apply (cases "r = 0") apply simp apply (cases n) apply auto doneqedtext \<open>Summability of geometric series for real algebras.\<close>lemma complete_algebra_summable_geometric: fixes x :: "'a::{real_normed_algebra_1,banach}" assumes "norm x < 1" shows "summable (\<lambda>n. x ^ n)"proof (rule summable_comparison_test) show "\<exists>N. \<forall>n\<ge>N. norm (x ^ n) \<le> norm x ^ n" by (simp add: norm_power_ineq) from assms show "summable (\<lambda>n. norm x ^ n)" by (simp add: summable_geometric)qedsubsection \<open>Cauchy Product Formula\<close>text \<open> Proof based on Analysis WebNotes: Chapter 07, Class 41 \<^url>\<open>http://www.math.unl.edu/~webnotes/classes/class41/prp77.htm\<close>\<close>lemma Cauchy_product_sums: fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}" assumes a: "summable (\<lambda>k. norm (a k))" and b: "summable (\<lambda>k. norm (b k))" shows "(\<lambda>k. \<Sum>i\<le>k. a i * b (k - i)) sums ((\<Sum>k. a k) * (\<Sum>k. b k))"proof - let ?S1 = "\<lambda>n::nat. {..<n} \<times> {..<n}" let ?S2 = "\<lambda>n::nat. {(i,j). i + j < n}" have S1_mono: "\<And>m n. m \<le> n \<Longrightarrow> ?S1 m \<subseteq> ?S1 n" by auto have S2_le_S1: "\<And>n. ?S2 n \<subseteq> ?S1 n" by auto have S1_le_S2: "\<And>n. ?S1 (n div 2) \<subseteq> ?S2 n" by auto have finite_S1: "\<And>n. finite (?S1 n)" by simp with S2_le_S1 have finite_S2: "\<And>n. finite (?S2 n)" by (rule finite_subset) let ?g = "\<lambda>(i,j). a i * b j" let ?f = "\<lambda>(i,j). norm (a i) * norm (b j)" have f_nonneg: "\<And>x. 0 \<le> ?f x" by auto then have norm_setsum_f: "\<And>A. norm (setsum ?f A) = setsum ?f A" unfolding real_norm_def by (simp only: abs_of_nonneg setsum_nonneg [rule_format]) have "(\<lambda>n. (\<Sum>k<n. a k) * (\<Sum>k<n. b k)) \<longlonglongrightarrow> (\<Sum>k. a k) * (\<Sum>k. b k)" by (intro tendsto_mult summable_LIMSEQ summable_norm_cancel [OF a] summable_norm_cancel [OF b]) then have 1: "(\<lambda>n. setsum ?g (?S1 n)) \<longlonglongrightarrow> (\<Sum>k. a k) * (\<Sum>k. b k)" by (simp only: setsum_product setsum.Sigma [rule_format] finite_lessThan) have "(\<lambda>n. (\<Sum>k<n. norm (a k)) * (\<Sum>k<n. norm (b k))) \<longlonglongrightarrow> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))" using a b by (intro tendsto_mult summable_LIMSEQ) then have "(\<lambda>n. setsum ?f (?S1 n)) \<longlonglongrightarrow> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))" by (simp only: setsum_product setsum.Sigma [rule_format] finite_lessThan) then have "convergent (\<lambda>n. setsum ?f (?S1 n))" by (rule convergentI) then have Cauchy: "Cauchy (\<lambda>n. setsum ?f (?S1 n))" by (rule convergent_Cauchy) have "Zfun (\<lambda>n. setsum ?f (?S1 n - ?S2 n)) sequentially" proof (rule ZfunI, simp only: eventually_sequentially norm_setsum_f) fix r :: real assume r: "0 < r" from CauchyD [OF Cauchy r] obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (setsum ?f (?S1 m) - setsum ?f (?S1 n)) < r" .. then have "\<And>m n. N \<le> n \<Longrightarrow> n \<le> m \<Longrightarrow> norm (setsum ?f (?S1 m - ?S1 n)) < r" by (simp only: setsum_diff finite_S1 S1_mono) then have N: "\<And>m n. N \<le> n \<Longrightarrow> n \<le> m \<Longrightarrow> setsum ?f (?S1 m - ?S1 n) < r" by (simp only: norm_setsum_f) show "\<exists>N. \<forall>n\<ge>N. setsum ?f (?S1 n - ?S2 n) < r" proof (intro exI allI impI) fix n assume "2 * N \<le> n" then have n: "N \<le> n div 2" by simp have "setsum ?f (?S1 n - ?S2 n) \<le> setsum ?f (?S1 n - ?S1 (n div 2))" by (intro setsum_mono2 finite_Diff finite_S1 f_nonneg Diff_mono subset_refl S1_le_S2) also have "\<dots> < r" using n div_le_dividend by (rule N) finally show "setsum ?f (?S1 n - ?S2 n) < r" . qed qed then have "Zfun (\<lambda>n. setsum ?g (?S1 n - ?S2 n)) sequentially" apply (rule Zfun_le [rule_format]) apply (simp only: norm_setsum_f) apply (rule order_trans [OF norm_setsum setsum_mono]) apply (auto simp add: norm_mult_ineq) done then have 2: "(\<lambda>n. setsum ?g (?S1 n) - setsum ?g (?S2 n)) \<longlonglongrightarrow> 0" unfolding tendsto_Zfun_iff diff_0_right by (simp only: setsum_diff finite_S1 S2_le_S1) with 1 have "(\<lambda>n. setsum ?g (?S2 n)) \<longlonglongrightarrow> (\<Sum>k. a k) * (\<Sum>k. b k)" by (rule Lim_transform2) then show ?thesis by (simp only: sums_def setsum_triangle_reindex)qedlemma Cauchy_product: fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}" assumes "summable (\<lambda>k. norm (a k))" and "summable (\<lambda>k. norm (b k))" shows "(\<Sum>k. a k) * (\<Sum>k. b k) = (\<Sum>k. \<Sum>i\<le>k. a i * b (k - i))" using assms by (rule Cauchy_product_sums [THEN sums_unique])lemma summable_Cauchy_product: fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}" assumes "summable (\<lambda>k. norm (a k))" and "summable (\<lambda>k. norm (b k))" shows "summable (\<lambda>k. \<Sum>i\<le>k. a i * b (k - i))" using Cauchy_product_sums[OF assms] by (simp add: sums_iff)subsection \<open>Series on @{typ real}s\<close>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))" by (rule summable_comparison_test) autolemma summable_rabs_comparison_test: "\<exists>N. \<forall>n\<ge>N. \<bar>f n\<bar> \<le> g n \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. \<bar>f n\<bar>)" for f :: "nat \<Rightarrow> real" by (rule summable_comparison_test) autolemma summable_rabs_cancel: "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> summable f" for f :: "nat \<Rightarrow> real" by (rule summable_norm_cancel) simplemma summable_rabs: "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> \<bar>suminf f\<bar> \<le> (\<Sum>n. \<bar>f n\<bar>)" for f :: "nat \<Rightarrow> real" by (fold real_norm_def) (rule summable_norm)lemma summable_zero_power [simp]: "summable (\<lambda>n. 0 ^ n :: 'a::{comm_ring_1,topological_space})"proof - have "(\<lambda>n. 0 ^ n :: 'a) = (\<lambda>n. if n = 0 then 0^0 else 0)" by (intro ext) (simp add: zero_power) moreover have "summable \<dots>" by simp ultimately show ?thesis by simpqedlemma summable_zero_power' [simp]: "summable (\<lambda>n. f n * 0 ^ n :: 'a::{ring_1,topological_space})"proof - have "(\<lambda>n. f n * 0 ^ n :: 'a) = (\<lambda>n. if n = 0 then f 0 * 0^0 else 0)" by (intro ext) (simp add: zero_power) moreover have "summable \<dots>" by simp ultimately show ?thesis by simpqedlemma summable_power_series: fixes z :: real assumes le_1: "\<And>i. f i \<le> 1" and nonneg: "\<And>i. 0 \<le> f i" and z: "0 \<le> z" "z < 1" shows "summable (\<lambda>i. f i * z^i)"proof (rule summable_comparison_test[OF _ summable_geometric]) show "norm z < 1" using z by (auto simp: less_imp_le) show "\<And>n. \<exists>N. \<forall>na\<ge>N. norm (f na * z ^ na) \<le> z ^ na" using z by (auto intro!: exI[of _ 0] mult_left_le_one_le simp: abs_mult nonneg power_abs less_imp_le le_1)qedlemma summable_0_powser: "summable (\<lambda>n. f n * 0 ^ n :: 'a::real_normed_div_algebra)"proof - have A: "(\<lambda>n. f n * 0 ^ n) = (\<lambda>n. if n = 0 then f n else 0)" by (intro ext) auto then show ?thesis by (subst A) simp_allqedlemma summable_powser_split_head: "summable (\<lambda>n. f (Suc n) * z ^ n :: 'a::real_normed_div_algebra) = summable (\<lambda>n. f n * z ^ n)"proof - have "summable (\<lambda>n. f (Suc n) * z ^ n) \<longleftrightarrow> summable (\<lambda>n. f (Suc n) * z ^ Suc n)" (is "?lhs \<longleftrightarrow> ?rhs") proof show ?rhs if ?lhs using summable_mult2[OF that, of z] by (simp add: power_commutes algebra_simps) show ?lhs if ?rhs using summable_mult2[OF that, of "inverse z"] by (cases "z \<noteq> 0", subst (asm) power_Suc2) (simp_all add: algebra_simps) qed also have "\<dots> \<longleftrightarrow> summable (\<lambda>n. f n * z ^ n)" by (rule summable_Suc_iff) finally show ?thesis .qedlemma powser_split_head: fixes f :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}" assumes "summable (\<lambda>n. f n * z ^ n)" shows "suminf (\<lambda>n. f n * z ^ n) = f 0 + suminf (\<lambda>n. f (Suc n) * z ^ n) * z" and "suminf (\<lambda>n. f (Suc n) * z ^ n) * z = suminf (\<lambda>n. f n * z ^ n) - f 0" and "summable (\<lambda>n. f (Suc n) * z ^ n)"proof - from assms show "summable (\<lambda>n. f (Suc n) * z ^ n)" by (subst summable_powser_split_head) from suminf_mult2[OF this, of z] have "(\<Sum>n. f (Suc n) * z ^ n) * z = (\<Sum>n. f (Suc n) * z ^ Suc n)" by (simp add: power_commutes algebra_simps) also from assms have "\<dots> = suminf (\<lambda>n. f n * z ^ n) - f 0" by (subst suminf_split_head) simp_all finally show "suminf (\<lambda>n. f n * z ^ n) = f 0 + suminf (\<lambda>n. f (Suc n) * z ^ n) * z" by simp then show "suminf (\<lambda>n. f (Suc n) * z ^ n) * z = suminf (\<lambda>n. f n * z ^ n) - f 0" by simpqedlemma summable_partial_sum_bound: fixes f :: "nat \<Rightarrow> 'a :: banach" and e :: real assumes summable: "summable f" and e: "e > 0" obtains N where "\<And>m n. m \<ge> N \<Longrightarrow> norm (\<Sum>k=m..n. f k) < e"proof - from summable have "Cauchy (\<lambda>n. \<Sum>k<n. f k)" by (simp add: Cauchy_convergent_iff summable_iff_convergent) from CauchyD [OF this e] obtain N where N: "\<And>m n. m \<ge> N \<Longrightarrow> n \<ge> N \<Longrightarrow> norm ((\<Sum>k<m. f k) - (\<Sum>k<n. f k)) < e" by blast have "norm (\<Sum>k=m..n. f k) < e" if m: "m \<ge> N" for m n proof (cases "n \<ge> m") case True with m have "norm ((\<Sum>k<Suc n. f k) - (\<Sum>k<m. f k)) < e" by (intro N) simp_all also from True have "(\<Sum>k<Suc n. f k) - (\<Sum>k<m. f k) = (\<Sum>k=m..n. f k)" by (subst setsum_diff [symmetric]) (simp_all add: setsum_last_plus) finally show ?thesis . next case False with e show ?thesis by simp_all qed then show ?thesis by (rule that)qedlemma powser_sums_if: "(\<lambda>n. (if n = m then (1 :: 'a::{ring_1,topological_space}) else 0) * z^n) sums z^m"proof - have "(\<lambda>n. (if n = m then 1 else 0) * z^n) = (\<lambda>n. if n = m then z^n else 0)" by (intro ext) auto then show ?thesis by (simp add: sums_single)qedlemma fixes f :: "nat \<Rightarrow> real" assumes "summable f" and "inj g" and pos: "\<And>x. 0 \<le> f x" shows summable_reindex: "summable (f \<circ> g)" and suminf_reindex_mono: "suminf (f \<circ> g) \<le> suminf f" and suminf_reindex: "(\<And>x. x \<notin> range g \<Longrightarrow> f x = 0) \<Longrightarrow> suminf (f \<circ> g) = suminf f"proof - from \<open>inj g\<close> have [simp]: "\<And>A. inj_on g A" by (rule subset_inj_on) simp have smaller: "\<forall>n. (\<Sum>i<n. (f \<circ> g) i) \<le> suminf f" proof fix n have "\<forall> n' \<in> (g ` {..<n}). n' < Suc (Max (g ` {..<n}))" by (metis Max_ge finite_imageI finite_lessThan not_le not_less_eq) then obtain m where n: "\<And>n'. n' < n \<Longrightarrow> g n' < m" by blast have "(\<Sum>i<n. f (g i)) = setsum f (g ` {..<n})" by (simp add: setsum.reindex) also have "\<dots> \<le> (\<Sum>i<m. f i)" by (rule setsum_mono3) (auto simp add: pos n[rule_format]) also have "\<dots> \<le> suminf f" using \<open>summable f\<close> by (rule setsum_le_suminf) (simp add: pos) finally show "(\<Sum>i<n. (f \<circ> g) i) \<le> suminf f" by simp qed have "incseq (\<lambda>n. \<Sum>i<n. (f \<circ> g) i)" by (rule incseq_SucI) (auto simp add: pos) then obtain L where L: "(\<lambda> n. \<Sum>i<n. (f \<circ> g) i) \<longlonglongrightarrow> L" using smaller by(rule incseq_convergent) then have "(f \<circ> g) sums L" by (simp add: sums_def) then show "summable (f \<circ> g)" by (auto simp add: sums_iff) then have "(\<lambda>n. \<Sum>i<n. (f \<circ> g) i) \<longlonglongrightarrow> suminf (f \<circ> g)" by (rule summable_LIMSEQ) then show le: "suminf (f \<circ> g) \<le> suminf f" by(rule LIMSEQ_le_const2)(blast intro: smaller[rule_format]) assume f: "\<And>x. x \<notin> range g \<Longrightarrow> f x = 0" from \<open>summable f\<close> have "suminf f \<le> suminf (f \<circ> g)" proof (rule suminf_le_const) fix n have "\<forall> n' \<in> (g -` {..<n}). n' < Suc (Max (g -` {..<n}))" by(auto intro: Max_ge simp add: finite_vimageI less_Suc_eq_le) then obtain m where n: "\<And>n'. g n' < n \<Longrightarrow> n' < m" by blast have "(\<Sum>i<n. f i) = (\<Sum>i\<in>{..<n} \<inter> range g. f i)" using f by(auto intro: setsum.mono_neutral_cong_right) also have "\<dots> = (\<Sum>i\<in>g -` {..<n}. (f \<circ> g) i)" by (rule setsum.reindex_cong[where l=g])(auto) also have "\<dots> \<le> (\<Sum>i<m. (f \<circ> g) i)" by (rule setsum_mono3)(auto simp add: pos n) also have "\<dots> \<le> suminf (f \<circ> g)" using \<open>summable (f \<circ> g)\<close> by (rule setsum_le_suminf) (simp add: pos) finally show "setsum f {..<n} \<le> suminf (f \<circ> g)" . qed with le show "suminf (f \<circ> g) = suminf f" by (rule antisym)qedlemma sums_mono_reindex: assumes subseq: "subseq g" and zero: "\<And>n. n \<notin> range g \<Longrightarrow> f n = 0" shows "(\<lambda>n. f (g n)) sums c \<longleftrightarrow> f sums c" unfolding sums_defproof assume lim: "(\<lambda>n. \<Sum>k<n. f k) \<longlonglongrightarrow> c" have "(\<lambda>n. \<Sum>k<n. f (g k)) = (\<lambda>n. \<Sum>k<g n. f k)" proof fix n :: nat from subseq have "(\<Sum>k<n. f (g k)) = (\<Sum>k\<in>g`{..<n}. f k)" by (subst setsum.reindex) (auto intro: subseq_imp_inj_on) also from subseq have "\<dots> = (\<Sum>k<g n. f k)" by (intro setsum.mono_neutral_left ballI zero) (auto dest: subseq_strict_mono simp: strict_mono_less strict_mono_less_eq) finally show "(\<Sum>k<n. f (g k)) = (\<Sum>k<g n. f k)" . qed also from LIMSEQ_subseq_LIMSEQ[OF lim subseq] have "\<dots> \<longlonglongrightarrow> c" by (simp only: o_def) finally show "(\<lambda>n. \<Sum>k<n. f (g k)) \<longlonglongrightarrow> c" .next assume lim: "(\<lambda>n. \<Sum>k<n. f (g k)) \<longlonglongrightarrow> c" define g_inv where "g_inv n = (LEAST m. g m \<ge> n)" for n from filterlim_subseq[OF subseq] have g_inv_ex: "\<exists>m. g m \<ge> n" for n by (auto simp: filterlim_at_top eventually_at_top_linorder) then have g_inv: "g (g_inv n) \<ge> n" for n unfolding g_inv_def by (rule LeastI_ex) have g_inv_least: "m \<ge> g_inv n" if "g m \<ge> n" for m n using that unfolding g_inv_def by (rule Least_le) have g_inv_least': "g m < n" if "m < g_inv n" for m n using that g_inv_least[of n m] by linarith have "(\<lambda>n. \<Sum>k<n. f k) = (\<lambda>n. \<Sum>k<g_inv n. f (g k))" proof fix n :: nat { fix k assume k: "k \<in> {..<n} - g`{..<g_inv n}" have "k \<notin> range g" proof (rule notI, elim imageE) fix l assume l: "k = g l" have "g l < g (g_inv n)" by (rule less_le_trans[OF _ g_inv]) (use k l in simp_all) with subseq have "l < g_inv n" by (simp add: subseq_strict_mono strict_mono_less) with k l show False by simp qed then have "f k = 0" by (rule zero) } with g_inv_least' g_inv have "(\<Sum>k<n. f k) = (\<Sum>k\<in>g`{..<g_inv n}. f k)" by (intro setsum.mono_neutral_right) auto also from subseq have "\<dots> = (\<Sum>k<g_inv n. f (g k))" using subseq_imp_inj_on by (subst setsum.reindex) simp_all finally show "(\<Sum>k<n. f k) = (\<Sum>k<g_inv n. f (g k))" . qed also { fix K n :: nat assume "g K \<le> n" also have "n \<le> g (g_inv n)" by (rule g_inv) finally have "K \<le> g_inv n" using subseq by (simp add: strict_mono_less_eq subseq_strict_mono) } then have "filterlim g_inv at_top sequentially" by (auto simp: filterlim_at_top eventually_at_top_linorder) with lim have "(\<lambda>n. \<Sum>k<g_inv n. f (g k)) \<longlonglongrightarrow> c" by (rule filterlim_compose) finally show "(\<lambda>n. \<Sum>k<n. f k) \<longlonglongrightarrow> c" .qedlemma summable_mono_reindex: assumes subseq: "subseq g" and zero: "\<And>n. n \<notin> range g \<Longrightarrow> f n = 0" shows "summable (\<lambda>n. f (g n)) \<longleftrightarrow> summable f" using sums_mono_reindex[of g f, OF assms] by (simp add: summable_def)lemma suminf_mono_reindex: fixes f :: "nat \<Rightarrow> 'a::{t2_space,comm_monoid_add}" assumes "subseq g" "\<And>n. n \<notin> range g \<Longrightarrow> f n = 0" shows "suminf (\<lambda>n. f (g n)) = suminf f"proof (cases "summable f") case True with sums_mono_reindex [of g f, OF assms] and summable_mono_reindex [of g f, OF assms] show ?thesis by (simp add: sums_iff)next case False then have "\<not>(\<exists>c. f sums c)" unfolding summable_def by blast then have "suminf f = The (\<lambda>_. False)" by (simp add: suminf_def) moreover from False have "\<not> summable (\<lambda>n. f (g n))" using summable_mono_reindex[of g f, OF assms] by simp then have "\<not>(\<exists>c. (\<lambda>n. f (g n)) sums c)" unfolding summable_def by blast then have "suminf (\<lambda>n. f (g n)) = The (\<lambda>_. False)" by (simp add: suminf_def) ultimately show ?thesis by simpqedend