author  huffman 
Tue, 17 Apr 2007 00:33:49 +0200  
changeset 22719  c51667189bd3 
parent 21404  eb85850d3eb7 
child 22852  2490d4b4671a 
permissions  rwrr 
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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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*) 
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header{*Finite Summation and Infinite Series*} 
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theory Series 
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imports SEQ 
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begin 
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definition 
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sums :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> bool" 
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(infixr "sums" 80) where 
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"f sums s = (%n. setsum f {0..<n}) > s" 
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definition 
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summable :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> bool" where 
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"summable f = (\<exists>s. f sums s)" 
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definition 
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suminf :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> 'a" where 
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"suminf f = (THE s. f sums s)" 
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syntax 
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"_suminf" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a" ("\<Sum>_. _" [0, 10] 10) 
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translations 
20770  32 
"\<Sum>i. b" == "CONST suminf (%i. b)" 
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lemma sumr_diff_mult_const: 
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"setsum f {0..<n}  (real n*r) = setsum (%i. f i  r) {0..<n::nat}" 

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by (simp add: diff_minus setsum_addf real_of_nat_def) 
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lemma real_setsum_nat_ivl_bounded: 
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"(!!p. p < n \<Longrightarrow> f(p) \<le> K) 

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\<Longrightarrow> setsum f {0..<n::nat} \<le> real n * K" 

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using setsum_bounded[where A = "{0..<n}"] 

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by (auto simp:real_of_nat_def) 

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(* Generalize from real to some algebraic structure? *) 
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lemma sumr_minus_one_realpow_zero [simp]: 

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"(\<Sum>i=0..<2*n. (1) ^ Suc i) = (0::real)" 
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by (induct "n", auto) 
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(* FIXME this is an awful lemma! *) 
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lemma sumr_one_lb_realpow_zero [simp]: 

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"(\<Sum>n=Suc 0..<n. f(n) * (0::real) ^ n) = 0" 

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by (rule setsum_0', simp) 
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lemma sumr_group: 
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"(\<Sum>m=0..<n::nat. setsum f {m * k ..< m*k + k}) = setsum f {0 ..< n * k}" 
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apply (subgoal_tac "k = 0  0 < k", auto) 
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apply (induct "n") 
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apply (simp_all add: setsum_add_nat_ivl add_commute) 
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done 
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lemma sumr_offset3: 
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"setsum f {0::nat..<n+k} = (\<Sum>m=0..<n. f (m+k)) + setsum f {0..<k}" 

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apply (subst setsum_shift_bounds_nat_ivl [symmetric]) 

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apply (simp add: setsum_add_nat_ivl add_commute) 

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done 

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lemma sumr_offset: 
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fixes f :: "nat \<Rightarrow> 'a::ab_group_add" 
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shows "(\<Sum>m=0..<n. f(m+k)) = setsum f {0..<n+k}  setsum f {0..<k}" 

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by (simp add: sumr_offset3) 

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lemma sumr_offset2: 

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"\<forall>f. (\<Sum>m=0..<n::nat. f(m+k)::real) = setsum f {0..<n+k}  setsum f {0..<k}" 

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by (simp add: sumr_offset) 
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lemma sumr_offset4: 

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"\<forall>n f. setsum f {0::nat..<n+k} = (\<Sum>m=0..<n. f (m+k)::real) + setsum f {0..<k}" 
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by (clarify, rule sumr_offset3) 

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(* 

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lemma sumr_from_1_from_0: "0 < n ==> 

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(\<Sum>n=Suc 0 ..< Suc n. if even(n) then 0 else 

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(( 1) ^ ((n  (Suc 0)) div 2))/(real (fact n))) * a ^ n = 

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(\<Sum>n=0..<Suc n. if even(n) then 0 else 

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(( 1) ^ ((n  (Suc 0)) div 2))/(real (fact n))) * a ^ n" 

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by (rule_tac n1 = 1 in sumr_split_add [THEN subst], auto) 

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*) 

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subsection{* Infinite Sums, by the Properties of Limits*} 

91 

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(* 

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suminf is the sum 

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*) 

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lemma sums_summable: "f sums l ==> summable f" 

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by (simp add: sums_def summable_def, blast) 

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lemma summable_sums: "summable f ==> f sums (suminf f)" 

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apply (simp add: summable_def suminf_def sums_def) 
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apply (blast intro: theI LIMSEQ_unique) 

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done 
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lemma summable_sumr_LIMSEQ_suminf: 

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"summable f ==> (%n. setsum f {0..<n}) > (suminf f)" 
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by (rule summable_sums [unfolded sums_def]) 
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(* 

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sum is unique 

109 
*) 

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lemma sums_unique: "f sums s ==> (s = suminf f)" 

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apply (frule sums_summable [THEN summable_sums]) 

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apply (auto intro!: LIMSEQ_unique simp add: sums_def) 

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done 

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lemma sums_split_initial_segment: "f sums s ==> 
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(%n. f(n + k)) sums (s  (SUM i = 0..< k. f i))" 

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apply (unfold sums_def); 

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apply (simp add: sumr_offset); 

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apply (rule LIMSEQ_diff_const) 

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apply (rule LIMSEQ_ignore_initial_segment) 

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apply assumption 

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done 

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lemma summable_ignore_initial_segment: "summable f ==> 

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summable (%n. f(n + k))" 

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apply (unfold summable_def) 

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apply (auto intro: sums_split_initial_segment) 

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done 

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lemma suminf_minus_initial_segment: "summable f ==> 

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suminf f = s ==> suminf (%n. f(n + k)) = s  (SUM i = 0..< k. f i)" 

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apply (frule summable_ignore_initial_segment) 

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apply (rule sums_unique [THEN sym]) 

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apply (frule summable_sums) 

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apply (rule sums_split_initial_segment) 

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apply auto 

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done 

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lemma suminf_split_initial_segment: "summable f ==> 

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suminf f = (SUM i = 0..< k. f i) + suminf (%n. f(n + k))" 

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by (auto simp add: suminf_minus_initial_segment) 

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lemma series_zero: 
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"(\<forall>m. n \<le> m > f(m) = 0) ==> f sums (setsum f {0..<n})" 
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apply (simp add: sums_def LIMSEQ_def diff_minus[symmetric], safe) 
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apply (rule_tac x = n in exI) 
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apply (clarsimp simp add:setsum_diff[symmetric] cong:setsum_ivl_cong) 
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done 
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lemma sums_zero: "(%n. 0) sums 0"; 
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apply (unfold sums_def); 

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apply simp; 

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apply (rule LIMSEQ_const); 

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done; 

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lemma summable_zero: "summable (%n. 0)"; 
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apply (rule sums_summable); 

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apply (rule sums_zero); 

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done; 

160 

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lemma suminf_zero: "suminf (%n. 0) = 0"; 

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apply (rule sym); 

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apply (rule sums_unique); 

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apply (rule sums_zero); 

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done; 

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lemma sums_mult: 
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fixes c :: "'a::real_normed_algebra" 

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shows "f sums a \<Longrightarrow> (\<lambda>n. c * f n) sums (c * a)" 

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by (auto simp add: sums_def setsum_right_distrib [symmetric] 
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intro!: LIMSEQ_mult intro: LIMSEQ_const) 
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lemma summable_mult: 
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fixes c :: "'a::real_normed_algebra" 

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shows "summable f \<Longrightarrow> summable (%n. c * f n)"; 

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apply (unfold summable_def); 
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apply (auto intro: sums_mult); 

178 
done; 

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lemma suminf_mult: 
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fixes c :: "'a::real_normed_algebra" 

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shows "summable f \<Longrightarrow> suminf (\<lambda>n. c * f n) = c * suminf f"; 

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apply (rule sym); 
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apply (rule sums_unique); 

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apply (rule sums_mult); 

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apply (erule summable_sums); 

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done; 

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lemma sums_mult2: 
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fixes c :: "'a::real_normed_algebra" 

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shows "f sums a \<Longrightarrow> (\<lambda>n. f n * c) sums (a * c)" 

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by (auto simp add: sums_def setsum_left_distrib [symmetric] 

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intro!: LIMSEQ_mult LIMSEQ_const) 

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lemma summable_mult2: 
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fixes c :: "'a::real_normed_algebra" 

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shows "summable f \<Longrightarrow> summable (\<lambda>n. f n * c)" 

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apply (unfold summable_def) 
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apply (auto intro: sums_mult2) 

200 
done 

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lemma suminf_mult2: 
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fixes c :: "'a::real_normed_algebra" 

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shows "summable f \<Longrightarrow> suminf f * c = (\<Sum>n. f n * c)" 

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by (auto intro!: sums_unique sums_mult2 summable_sums) 

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lemma sums_divide: 
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fixes c :: "'a::real_normed_field" 

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shows "f sums a \<Longrightarrow> (\<lambda>n. f n / c) sums (a / c)" 

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by (simp add: divide_inverse sums_mult2) 

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lemma summable_divide: 
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fixes c :: "'a::real_normed_field" 

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shows "summable f \<Longrightarrow> summable (\<lambda>n. f n / c)" 

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apply (unfold summable_def); 
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apply (auto intro: sums_divide); 

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done; 

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lemma suminf_divide: 
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fixes c :: "'a::real_normed_field" 

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shows "summable f \<Longrightarrow> suminf (\<lambda>n. f n / c) = suminf f / c" 

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apply (rule sym); 
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apply (rule sums_unique); 

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apply (rule sums_divide); 

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apply (erule summable_sums); 

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done; 

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lemma sums_add: "[ x sums x0; y sums y0 ] ==> (%n. x n + y n) sums (x0+y0)" 

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by (auto simp add: sums_def setsum_addf intro: LIMSEQ_add) 

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lemma summable_add: "summable f ==> summable g ==> summable (%x. f x + g x)"; 

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apply (unfold summable_def); 

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apply clarify; 

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apply (rule exI); 

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apply (erule sums_add); 

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apply assumption; 

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done; 

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lemma suminf_add: 

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"[ summable f; summable g ] 

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==> suminf f + suminf g = (\<Sum>n. f n + g n)" 

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by (auto intro!: sums_add sums_unique summable_sums) 

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lemma sums_diff: "[ x sums x0; y sums y0 ] ==> (%n. x n  y n) sums (x0y0)" 
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by (auto simp add: sums_def setsum_subtractf intro: LIMSEQ_diff) 
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lemma summable_diff: "summable f ==> summable g ==> summable (%x. f x  g x)"; 
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apply (unfold summable_def); 

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apply clarify; 

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apply (rule exI); 

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apply (erule sums_diff); 

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apply assumption; 

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done; 

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lemma suminf_diff: 

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"[ summable f; summable g ] 

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==> suminf f  suminf g = (\<Sum>n. f n  g n)" 
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by (auto intro!: sums_diff sums_unique summable_sums) 
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lemma sums_minus: "f sums s ==> (%x.  f x) sums ( s)"; 
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by (simp add: sums_def setsum_negf LIMSEQ_minus); 

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lemma summable_minus: "summable f ==> summable (%x.  f x)"; 

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by (auto simp add: summable_def intro: sums_minus); 

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lemma suminf_minus: "summable f ==> suminf (%x.  f x) =  (suminf f)"; 

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apply (rule sym); 

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apply (rule sums_unique); 

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apply (rule sums_minus); 

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apply (erule summable_sums); 

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done; 

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lemma sums_group: 

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"[summable f; 0 < k ] ==> (%n. setsum f {n*k..<n*k+k}) sums (suminf f)" 
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apply (drule summable_sums) 
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apply (simp only: sums_def sumr_group) 
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apply (unfold LIMSEQ_def, safe) 

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apply (drule_tac x="r" in spec, safe) 

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apply (rule_tac x="no" in exI, safe) 

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

281 
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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text{*A summable series of positive terms has limit that is at least as 
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great as any partial sum.*} 
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lemma series_pos_le: 
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fixes f :: "nat \<Rightarrow> real" 

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shows "\<lbrakk>summable f; \<forall>m\<ge>n. 0 \<le> f m\<rbrakk> \<Longrightarrow> setsum f {0..<n} \<le> suminf f" 

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apply (drule summable_sums) 
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apply (simp add: sums_def) 

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apply (cut_tac k = "setsum f {0..<n}" in LIMSEQ_const) 
295 
apply (erule LIMSEQ_le, blast) 

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apply (rule_tac x="n" in exI, clarify) 
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apply (rule setsum_mono2) 
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apply auto 

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done 
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301 
lemma series_pos_less: 

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fixes f :: "nat \<Rightarrow> real" 
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shows "\<lbrakk>summable f; \<forall>m\<ge>n. 0 < f m\<rbrakk> \<Longrightarrow> setsum f {0..<n} < suminf f" 

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apply (rule_tac y="setsum f {0..<Suc n}" in order_less_le_trans) 

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apply simp 

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apply (erule series_pos_le) 

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apply (simp add: order_less_imp_le) 

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done 

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lemma suminf_gt_zero: 

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fixes f :: "nat \<Rightarrow> real" 

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shows "\<lbrakk>summable f; \<forall>n. 0 < f n\<rbrakk> \<Longrightarrow> 0 < suminf f" 

313 
by (drule_tac n="0" in series_pos_less, simp_all) 

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315 
lemma suminf_ge_zero: 

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fixes f :: "nat \<Rightarrow> real" 

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shows "\<lbrakk>summable f; \<forall>n. 0 \<le> f n\<rbrakk> \<Longrightarrow> 0 \<le> suminf f" 

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by (drule_tac n="0" in series_pos_le, simp_all) 

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320 
lemma sumr_pos_lt_pair: 

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fixes f :: "nat \<Rightarrow> real" 

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shows "\<lbrakk>summable f; 

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\<forall>d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc(Suc 0) * d) + 1))\<rbrakk> 

324 
\<Longrightarrow> setsum f {0..<k} < suminf f" 

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apply (subst suminf_split_initial_segment [where k="k"]) 

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apply assumption 

327 
apply simp 

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apply (drule_tac k="k" in summable_ignore_initial_segment) 

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apply (drule_tac k="Suc (Suc 0)" in sums_group, simp) 

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apply simp 

331 
apply (frule sums_unique) 

332 
apply (drule sums_summable) 

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apply simp 

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apply (erule suminf_gt_zero) 

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apply (simp add: add_ac) 

14416  336 
done 
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text{*Sum of a geometric progression.*} 
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lemmas sumr_geometric = geometric_sum [where 'a = real] 
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lemma geometric_sums: 
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fixes x :: "'a::{real_normed_field,recpower}" 
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shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) sums (1 / (1  x))" 
345 
proof  

346 
assume less_1: "norm x < 1" 

347 
hence neq_1: "x \<noteq> 1" by auto 

348 
hence neq_0: "x  1 \<noteq> 0" by simp 

349 
from less_1 have lim_0: "(\<lambda>n. x ^ n) > 0" 

350 
by (rule LIMSEQ_power_zero) 

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hence "(\<lambda>n. x ^ n / (x  1)  1 / (x  1)) > 0 / (x  1)  1 / (x  1)" 
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using neq_0 by (intro LIMSEQ_divide LIMSEQ_diff LIMSEQ_const) 
353 
hence "(\<lambda>n. (x ^ n  1) / (x  1)) > 1 / (1  x)" 

354 
by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib) 

355 
thus "(\<lambda>n. x ^ n) sums (1 / (1  x))" 

356 
by (simp add: sums_def geometric_sum neq_1) 

357 
qed 

358 

359 
lemma summable_geometric: 

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fixes x :: "'a::{real_normed_field,recpower}" 
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shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)" 
362 
by (rule geometric_sums [THEN sums_summable]) 

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text{*Cauchytype criterion for convergence of series (c.f. Harrison)*} 
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15539  366 
lemma summable_convergent_sumr_iff: 
367 
"summable f = convergent (%n. setsum f {0..<n})" 

14416  368 
by (simp add: summable_def sums_def convergent_def) 
369 

20689  370 
lemma summable_LIMSEQ_zero: "summable f \<Longrightarrow> f > 0" 
371 
apply (drule summable_convergent_sumr_iff [THEN iffD1]) 

20692  372 
apply (drule convergent_Cauchy) 
20689  373 
apply (simp only: Cauchy_def LIMSEQ_def, safe) 
374 
apply (drule_tac x="r" in spec, safe) 

375 
apply (rule_tac x="M" in exI, safe) 

376 
apply (drule_tac x="Suc n" in spec, simp) 

377 
apply (drule_tac x="n" in spec, simp) 

378 
done 

379 

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lemma summable_Cauchy: 
20848  381 
"summable (f::nat \<Rightarrow> 'a::banach) = 
382 
(\<forall>e > 0. \<exists>N. \<forall>m \<ge> N. \<forall>n. norm (setsum f {m..<n}) < e)" 

383 
apply (simp only: summable_convergent_sumr_iff Cauchy_convergent_iff [symmetric] Cauchy_def, safe) 

20410  384 
apply (drule spec, drule (1) mp) 
385 
apply (erule exE, rule_tac x="M" in exI, clarify) 

386 
apply (rule_tac x="m" and y="n" in linorder_le_cases) 

387 
apply (frule (1) order_trans) 

388 
apply (drule_tac x="n" in spec, drule (1) mp) 

389 
apply (drule_tac x="m" in spec, drule (1) mp) 

390 
apply (simp add: setsum_diff [symmetric]) 

391 
apply simp 

392 
apply (drule spec, drule (1) mp) 

393 
apply (erule exE, rule_tac x="N" in exI, clarify) 

394 
apply (rule_tac x="m" and y="n" in linorder_le_cases) 

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apply (subst norm_minus_commute) 
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apply (simp add: setsum_diff [symmetric]) 
397 
apply (simp add: setsum_diff [symmetric]) 

14416  398 
done 
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text{*Comparison test*} 
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20692  402 
lemma norm_setsum: 
403 
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" 

404 
shows "norm (setsum f A) \<le> (\<Sum>i\<in>A. norm (f i))" 

405 
apply (case_tac "finite A") 

406 
apply (erule finite_induct) 

407 
apply simp 

408 
apply simp 

409 
apply (erule order_trans [OF norm_triangle_ineq add_left_mono]) 

410 
apply simp 

411 
done 

412 

14416  413 
lemma summable_comparison_test: 
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fixes f :: "nat \<Rightarrow> 'a::banach" 
415 
shows "\<lbrakk>\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable f" 

20692  416 
apply (simp add: summable_Cauchy, safe) 
417 
apply (drule_tac x="e" in spec, safe) 

418 
apply (rule_tac x = "N + Na" in exI, safe) 

14416  419 
apply (rotate_tac 2) 
420 
apply (drule_tac x = m in spec) 

421 
apply (auto, rotate_tac 2, drule_tac x = n in spec) 

20848  422 
apply (rule_tac y = "\<Sum>k=m..<n. norm (f k)" in order_le_less_trans) 
423 
apply (rule norm_setsum) 

15539  424 
apply (rule_tac y = "setsum g {m..<n}" in order_le_less_trans) 
425 
apply (auto intro: setsum_mono simp add: abs_interval_iff) 

14416  426 
done 
427 

20848  428 
lemma summable_norm_comparison_test: 
429 
fixes f :: "nat \<Rightarrow> 'a::banach" 

430 
shows "\<lbrakk>\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n; summable g\<rbrakk> 

431 
\<Longrightarrow> summable (\<lambda>n. norm (f n))" 

432 
apply (rule summable_comparison_test) 

433 
apply (auto) 

434 
done 

435 

14416  436 
lemma summable_rabs_comparison_test: 
20692  437 
fixes f :: "nat \<Rightarrow> real" 
438 
shows "\<lbrakk>\<exists>N. \<forall>n\<ge>N. \<bar>f n\<bar> \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable (\<lambda>n. \<bar>f n\<bar>)" 

14416  439 
apply (rule summable_comparison_test) 
15543  440 
apply (auto) 
14416  441 
done 
442 

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443 
text{*Limit comparison property for series (c.f. jrh)*} 
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444 

14416  445 
lemma summable_le: 
20692  446 
fixes f g :: "nat \<Rightarrow> real" 
447 
shows "\<lbrakk>\<forall>n. f n \<le> g n; summable f; summable g\<rbrakk> \<Longrightarrow> suminf f \<le> suminf g" 

14416  448 
apply (drule summable_sums)+ 
20692  449 
apply (simp only: sums_def, erule (1) LIMSEQ_le) 
14416  450 
apply (rule exI) 
15539  451 
apply (auto intro!: setsum_mono) 
14416  452 
done 
453 

454 
lemma summable_le2: 

20692  455 
fixes f g :: "nat \<Rightarrow> real" 
456 
shows "\<lbrakk>\<forall>n. \<bar>f n\<bar> \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable f \<and> suminf f \<le> suminf g" 

20848  457 
apply (subgoal_tac "summable f") 
458 
apply (auto intro!: summable_le) 

14416  459 
apply (simp add: abs_le_interval_iff) 
20848  460 
apply (rule_tac g="g" in summable_comparison_test, simp_all) 
14416  461 
done 
462 

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463 
(* specialisation for the common 0 case *) 
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464 
lemma suminf_0_le: 
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465 
fixes f::"nat\<Rightarrow>real" 
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466 
assumes gt0: "\<forall>n. 0 \<le> f n" and sm: "summable f" 
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467 
shows "0 \<le> suminf f" 
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468 
proof  
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469 
let ?g = "(\<lambda>n. (0::real))" 
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470 
from gt0 have "\<forall>n. ?g n \<le> f n" by simp 
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471 
moreover have "summable ?g" by (rule summable_zero) 
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472 
moreover from sm have "summable f" . 
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473 
ultimately have "suminf ?g \<le> suminf f" by (rule summable_le) 
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474 
then show "0 \<le> suminf f" by (simp add: suminf_zero) 
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475 
qed 
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476 

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477 

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478 
text{*Absolute convergence imples normal convergence*} 
20848  479 
lemma summable_norm_cancel: 
480 
fixes f :: "nat \<Rightarrow> 'a::banach" 

481 
shows "summable (\<lambda>n. norm (f n)) \<Longrightarrow> summable f" 

20692  482 
apply (simp only: summable_Cauchy, safe) 
483 
apply (drule_tac x="e" in spec, safe) 

484 
apply (rule_tac x="N" in exI, safe) 

485 
apply (drule_tac x="m" in spec, safe) 

20848  486 
apply (rule order_le_less_trans [OF norm_setsum]) 
487 
apply (rule order_le_less_trans [OF abs_ge_self]) 

20692  488 
apply simp 
14416  489 
done 
490 

20848  491 
lemma summable_rabs_cancel: 
492 
fixes f :: "nat \<Rightarrow> real" 

493 
shows "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> summable f" 

494 
by (rule summable_norm_cancel, simp) 

495 

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496 
text{*Absolute convergence of series*} 
20848  497 
lemma summable_norm: 
498 
fixes f :: "nat \<Rightarrow> 'a::banach" 

499 
shows "summable (\<lambda>n. norm (f n)) \<Longrightarrow> norm (suminf f) \<le> (\<Sum>n. norm (f n))" 

500 
by (auto intro: LIMSEQ_le LIMSEQ_norm summable_norm_cancel 

501 
summable_sumr_LIMSEQ_suminf norm_setsum) 

502 

14416  503 
lemma summable_rabs: 
20692  504 
fixes f :: "nat \<Rightarrow> real" 
505 
shows "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> \<bar>suminf f\<bar> \<le> (\<Sum>n. \<bar>f n\<bar>)" 

20848  506 
by (fold real_norm_def, rule summable_norm) 
14416  507 

508 
subsection{* The Ratio Test*} 

509 

20848  510 
lemma norm_ratiotest_lemma: 
511 
fixes x y :: "'a::normed" 

512 
shows "\<lbrakk>c \<le> 0; norm x \<le> c * norm y\<rbrakk> \<Longrightarrow> x = 0" 

513 
apply (subgoal_tac "norm x \<le> 0", simp) 

514 
apply (erule order_trans) 

515 
apply (simp add: mult_le_0_iff) 

516 
done 

517 

14416  518 
lemma rabs_ratiotest_lemma: "[ c \<le> 0; abs x \<le> c * abs y ] ==> x = (0::real)" 
20848  519 
by (erule norm_ratiotest_lemma, simp) 
14416  520 

521 
lemma le_Suc_ex: "(k::nat) \<le> l ==> (\<exists>n. l = k + n)" 

522 
apply (drule le_imp_less_or_eq) 

523 
apply (auto dest: less_imp_Suc_add) 

524 
done 

525 

526 
lemma le_Suc_ex_iff: "((k::nat) \<le> l) = (\<exists>n. l = k + n)" 

527 
by (auto simp add: le_Suc_ex) 

528 

529 
(*All this trouble just to get 0<c *) 

530 
lemma ratio_test_lemma2: 

20848  531 
fixes f :: "nat \<Rightarrow> 'a::banach" 
532 
shows "\<lbrakk>\<forall>n\<ge>N. norm (f (Suc n)) \<le> c * norm (f n)\<rbrakk> \<Longrightarrow> 0 < c \<or> summable f" 

14416  533 
apply (simp (no_asm) add: linorder_not_le [symmetric]) 
534 
apply (simp add: summable_Cauchy) 

15543  535 
apply (safe, subgoal_tac "\<forall>n. N < n > f (n) = 0") 
536 
prefer 2 

537 
apply clarify 

538 
apply(erule_tac x = "n  1" in allE) 

539 
apply (simp add:diff_Suc split:nat.splits) 

20848  540 
apply (blast intro: norm_ratiotest_lemma) 
14416  541 
apply (rule_tac x = "Suc N" in exI, clarify) 
15543  542 
apply(simp cong:setsum_ivl_cong) 
14416  543 
done 
544 

545 
lemma ratio_test: 

20848  546 
fixes f :: "nat \<Rightarrow> 'a::banach" 
547 
shows "\<lbrakk>c < 1; \<forall>n\<ge>N. norm (f (Suc n)) \<le> c * norm (f n)\<rbrakk> \<Longrightarrow> summable f" 

14416  548 
apply (frule ratio_test_lemma2, auto) 
20848  549 
apply (rule_tac g = "%n. (norm (f N) / (c ^ N))*c ^ n" 
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550 
in summable_comparison_test) 
14416  551 
apply (rule_tac x = N in exI, safe) 
552 
apply (drule le_Suc_ex_iff [THEN iffD1]) 

553 
apply (auto simp add: power_add realpow_not_zero) 

15539  554 
apply (induct_tac "na", auto) 
20848  555 
apply (rule_tac y = "c * norm (f (N + n))" in order_trans) 
14416  556 
apply (auto intro: mult_right_mono simp add: summable_def) 
557 
apply (simp add: mult_ac) 

20848  558 
apply (rule_tac x = "norm (f N) * (1/ (1  c)) / (c ^ N)" in exI) 
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559 
apply (rule sums_divide) 
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560 
apply (rule sums_mult) 
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561 
apply (auto intro!: geometric_sums) 
14416  562 
done 
563 

564 
end 