src/HOL/Analysis/Summation_Tests.thy
 author wenzelm Sun Sep 18 20:33:48 2016 +0200 (2016-09-18) changeset 63915 bab633745c7f parent 63627 6ddb43c6b711 child 63918 6bf55e6e0b75 permissions -rw-r--r--
tuned proofs;
```     1 (*  Title:    HOL/Analysis/Summation.thy
```
```     2     Author:   Manuel Eberl, TU München
```
```     3 *)
```
```     4
```
```     5 section \<open>Radius of Convergence and Summation Tests\<close>
```
```     6
```
```     7 theory Summation_Tests
```
```     8 imports
```
```     9   Complex_Main
```
```    10   "~~/src/HOL/Library/Extended_Real"
```
```    11   "~~/src/HOL/Library/Liminf_Limsup"
```
```    12 begin
```
```    13
```
```    14 text \<open>
```
```    15   The definition of the radius of convergence of a power series,
```
```    16   various summability tests, lemmas to compute the radius of convergence etc.
```
```    17 \<close>
```
```    18
```
```    19 subsection \<open>Rounded dual logarithm\<close>
```
```    20
```
```    21 (* This is required for the Cauchy condensation criterion *)
```
```    22
```
```    23 definition "natlog2 n = (if n = 0 then 0 else nat \<lfloor>log 2 (real_of_nat n)\<rfloor>)"
```
```    24
```
```    25 lemma natlog2_0 [simp]: "natlog2 0 = 0" by (simp add: natlog2_def)
```
```    26 lemma natlog2_1 [simp]: "natlog2 1 = 0" by (simp add: natlog2_def)
```
```    27 lemma natlog2_eq_0_iff: "natlog2 n = 0 \<longleftrightarrow> n < 2" by (simp add: natlog2_def)
```
```    28
```
```    29 lemma natlog2_power_of_two [simp]: "natlog2 (2 ^ n) = n"
```
```    30   by (simp add: natlog2_def log_nat_power)
```
```    31
```
```    32 lemma natlog2_mono: "m \<le> n \<Longrightarrow> natlog2 m \<le> natlog2 n"
```
```    33   unfolding natlog2_def by (simp_all add: nat_mono floor_mono)
```
```    34
```
```    35 lemma pow_natlog2_le: "n > 0 \<Longrightarrow> 2 ^ natlog2 n \<le> n"
```
```    36 proof -
```
```    37   assume n: "n > 0"
```
```    38   from n have "of_nat (2 ^ natlog2 n) = 2 powr real_of_nat (nat \<lfloor>log 2 (real_of_nat n)\<rfloor>)"
```
```    39     by (subst powr_realpow) (simp_all add: natlog2_def)
```
```    40   also have "\<dots> = 2 powr of_int \<lfloor>log 2 (real_of_nat n)\<rfloor>" using n by simp
```
```    41   also have "\<dots> \<le> 2 powr log 2 (real_of_nat n)" by (intro powr_mono) (linarith, simp_all)
```
```    42   also have "\<dots> = of_nat n" using n by simp
```
```    43   finally show ?thesis by simp
```
```    44 qed
```
```    45
```
```    46 lemma pow_natlog2_gt: "n > 0 \<Longrightarrow> 2 * 2 ^ natlog2 n > n"
```
```    47   and pow_natlog2_ge: "n > 0 \<Longrightarrow> 2 * 2 ^ natlog2 n \<ge> n"
```
```    48 proof -
```
```    49   assume n: "n > 0"
```
```    50   from n have "of_nat n = 2 powr log 2 (real_of_nat n)" by simp
```
```    51   also have "\<dots> < 2 powr (1 + of_int \<lfloor>log 2 (real_of_nat n)\<rfloor>)"
```
```    52     by (intro powr_less_mono) (linarith, simp_all)
```
```    53   also from n have "\<dots> = 2 powr (1 + real_of_nat (nat \<lfloor>log 2 (real_of_nat n)\<rfloor>))" by simp
```
```    54   also from n have "\<dots> = of_nat (2 * 2 ^ natlog2 n)"
```
```    55     by (simp_all add: natlog2_def powr_real_of_int powr_add)
```
```    56   finally show "2 * 2 ^ natlog2 n > n" by (rule of_nat_less_imp_less)
```
```    57   thus "2 * 2 ^ natlog2 n \<ge> n" by simp
```
```    58 qed
```
```    59
```
```    60 lemma natlog2_eqI:
```
```    61   assumes "n > 0" "2^k \<le> n" "n < 2 * 2^k"
```
```    62   shows   "natlog2 n = k"
```
```    63 proof -
```
```    64   from assms have "of_nat (2 ^ k) \<le> real_of_nat n"  by (subst of_nat_le_iff) simp_all
```
```    65   hence "real_of_int (int k) \<le> log (of_nat 2) (real_of_nat n)"
```
```    66     by (subst le_log_iff) (simp_all add: powr_realpow assms del: of_nat_le_iff)
```
```    67   moreover from assms have "real_of_nat n < of_nat (2 ^ Suc k)" by (subst of_nat_less_iff) simp_all
```
```    68   hence "log 2 (real_of_nat n) < of_nat k + 1"
```
```    69     by (subst log_less_iff) (simp_all add: assms powr_realpow powr_add)
```
```    70   ultimately have "\<lfloor>log 2 (real_of_nat n)\<rfloor> = of_nat k" by (intro floor_unique) simp_all
```
```    71   with assms show ?thesis by (simp add: natlog2_def)
```
```    72 qed
```
```    73
```
```    74 lemma natlog2_rec:
```
```    75   assumes "n \<ge> 2"
```
```    76   shows   "natlog2 n = 1 + natlog2 (n div 2)"
```
```    77 proof (rule natlog2_eqI)
```
```    78   from assms have "2 ^ (1 + natlog2 (n div 2)) \<le> 2 * (n div 2)"
```
```    79     by (simp add: pow_natlog2_le)
```
```    80   also have "\<dots> \<le> n" by simp
```
```    81   finally show "2 ^ (1 + natlog2 (n div 2)) \<le> n" .
```
```    82 next
```
```    83   from assms have "n < 2 * (n div 2 + 1)" by simp
```
```    84   also from assms have "(n div 2) < 2 ^ (1 + natlog2 (n div 2))"
```
```    85     by (simp add: pow_natlog2_gt)
```
```    86   hence "2 * (n div 2 + 1) \<le> 2 * (2 ^ (1 + natlog2 (n div 2)))"
```
```    87     by (intro mult_left_mono) simp_all
```
```    88   finally show "n < 2 * 2 ^ (1 + natlog2 (n div 2))" .
```
```    89 qed (insert assms, simp_all)
```
```    90
```
```    91 fun natlog2_aux where
```
```    92   "natlog2_aux n acc = (if (n::nat) < 2 then acc else natlog2_aux (n div 2) (acc + 1))"
```
```    93
```
```    94 lemma natlog2_aux_correct:
```
```    95   "natlog2_aux n acc = acc + natlog2 n"
```
```    96   by (induction n acc rule: natlog2_aux.induct) (auto simp: natlog2_rec natlog2_eq_0_iff)
```
```    97
```
```    98 lemma natlog2_code [code]: "natlog2 n = natlog2_aux n 0"
```
```    99   by (subst natlog2_aux_correct) simp
```
```   100
```
```   101
```
```   102 subsection \<open>Convergence tests for infinite sums\<close>
```
```   103
```
```   104 subsubsection \<open>Root test\<close>
```
```   105
```
```   106 lemma limsup_root_powser:
```
```   107   fixes f :: "nat \<Rightarrow> 'a :: {banach, real_normed_div_algebra}"
```
```   108   shows "limsup (\<lambda>n. ereal (root n (norm (f n * z ^ n)))) =
```
```   109              limsup (\<lambda>n. ereal (root n (norm (f n)))) * ereal (norm z)"
```
```   110 proof -
```
```   111   have A: "(\<lambda>n. ereal (root n (norm (f n * z ^ n)))) =
```
```   112               (\<lambda>n. ereal (root n (norm (f n))) * ereal (norm z))" (is "?g = ?h")
```
```   113   proof
```
```   114     fix n show "?g n = ?h n"
```
```   115     by (cases "n = 0") (simp_all add: norm_mult real_root_mult real_root_pos2 norm_power)
```
```   116   qed
```
```   117   show ?thesis by (subst A, subst limsup_ereal_mult_right) simp_all
```
```   118 qed
```
```   119
```
```   120 lemma limsup_root_limit:
```
```   121   assumes "(\<lambda>n. ereal (root n (norm (f n)))) \<longlonglongrightarrow> l" (is "?g \<longlonglongrightarrow> _")
```
```   122   shows   "limsup (\<lambda>n. ereal (root n (norm (f n)))) = l"
```
```   123 proof -
```
```   124   from assms have "convergent ?g" "lim ?g = l"
```
```   125     unfolding convergent_def by (blast intro: limI)+
```
```   126   with convergent_limsup_cl show ?thesis by force
```
```   127 qed
```
```   128
```
```   129 lemma limsup_root_limit':
```
```   130   assumes "(\<lambda>n. root n (norm (f n))) \<longlonglongrightarrow> l"
```
```   131   shows   "limsup (\<lambda>n. ereal (root n (norm (f n)))) = ereal l"
```
```   132   by (intro limsup_root_limit tendsto_ereal assms)
```
```   133
```
```   134 lemma root_test_convergence':
```
```   135   fixes f :: "nat \<Rightarrow> 'a :: banach"
```
```   136   defines "l \<equiv> limsup (\<lambda>n. ereal (root n (norm (f n))))"
```
```   137   assumes l: "l < 1"
```
```   138   shows   "summable f"
```
```   139 proof -
```
```   140   have "0 = limsup (\<lambda>n. 0)" by (simp add: Limsup_const)
```
```   141   also have "... \<le> l" unfolding l_def by (intro Limsup_mono) (simp_all add: real_root_ge_zero)
```
```   142   finally have "l \<ge> 0" by simp
```
```   143   with l obtain l' where l': "l = ereal l'" by (cases l) simp_all
```
```   144
```
```   145   define c where "c = (1 - l') / 2"
```
```   146   from l and \<open>l \<ge> 0\<close> have c: "l + c > l" "l' + c \<ge> 0" "l' + c < 1" unfolding c_def
```
```   147     by (simp_all add: field_simps l')
```
```   148   have "\<forall>C>l. eventually (\<lambda>n. ereal (root n (norm (f n))) < C) sequentially"
```
```   149     by (subst Limsup_le_iff[symmetric]) (simp add: l_def)
```
```   150   with c have "eventually (\<lambda>n. ereal (root n (norm (f n))) < l + ereal c) sequentially" by simp
```
```   151   with eventually_gt_at_top[of "0::nat"]
```
```   152     have "eventually (\<lambda>n. norm (f n) \<le> (l' + c) ^ n) sequentially"
```
```   153   proof eventually_elim
```
```   154     fix n :: nat assume n: "n > 0"
```
```   155     assume "ereal (root n (norm (f n))) < l + ereal c"
```
```   156     hence "root n (norm (f n)) \<le> l' + c" by (simp add: l')
```
```   157     with c n have "root n (norm (f n)) ^ n \<le> (l' + c) ^ n"
```
```   158       by (intro power_mono) (simp_all add: real_root_ge_zero)
```
```   159     also from n have "root n (norm (f n)) ^ n = norm (f n)" by simp
```
```   160     finally show "norm (f n) \<le> (l' + c) ^ n" by simp
```
```   161   qed
```
```   162   thus ?thesis
```
```   163     by (rule summable_comparison_test_ev[OF _ summable_geometric]) (simp add: c)
```
```   164 qed
```
```   165
```
```   166 lemma root_test_divergence:
```
```   167   fixes f :: "nat \<Rightarrow> 'a :: banach"
```
```   168   defines "l \<equiv> limsup (\<lambda>n. ereal (root n (norm (f n))))"
```
```   169   assumes l: "l > 1"
```
```   170   shows   "\<not>summable f"
```
```   171 proof
```
```   172   assume "summable f"
```
```   173   hence bounded: "Bseq f" by (simp add: summable_imp_Bseq)
```
```   174
```
```   175   have "0 = limsup (\<lambda>n. 0)" by (simp add: Limsup_const)
```
```   176   also have "... \<le> l" unfolding l_def by (intro Limsup_mono) (simp_all add: real_root_ge_zero)
```
```   177   finally have l_nonneg: "l \<ge> 0" by simp
```
```   178
```
```   179   define c where "c = (if l = \<infinity> then 2 else 1 + (real_of_ereal l - 1) / 2)"
```
```   180   from l l_nonneg consider "l = \<infinity>" | "\<exists>l'. l = ereal l'" by (cases l) simp_all
```
```   181   hence c: "c > 1 \<and> ereal c < l" by cases (insert l, auto simp: c_def field_simps)
```
```   182
```
```   183   have unbounded: "\<not>bdd_above {n. root n (norm (f n)) > c}"
```
```   184   proof
```
```   185     assume "bdd_above {n. root n (norm (f n)) > c}"
```
```   186     then obtain N where "\<forall>n. root n (norm (f n)) > c \<longrightarrow> n \<le> N" unfolding bdd_above_def by blast
```
```   187     hence "\<exists>N. \<forall>n\<ge>N. root n (norm (f n)) \<le> c"
```
```   188       by (intro exI[of _ "N + 1"]) (force simp: not_less_eq_eq[symmetric])
```
```   189     hence "eventually (\<lambda>n. root n (norm (f n)) \<le> c) sequentially"
```
```   190       by (auto simp: eventually_at_top_linorder)
```
```   191     hence "l \<le> c" unfolding l_def by (intro Limsup_bounded) simp_all
```
```   192     with c show False by auto
```
```   193   qed
```
```   194
```
```   195   from bounded obtain K where K: "K > 0" "\<And>n. norm (f n) \<le> K" using BseqE by blast
```
```   196   define n where "n = nat \<lceil>log c K\<rceil>"
```
```   197   from unbounded have "\<exists>m>n. c < root m (norm (f m))" unfolding bdd_above_def
```
```   198     by (auto simp: not_le)
```
```   199   then guess m by (elim exE conjE) note m = this
```
```   200   from c K have "K = c powr log c K" by (simp add: powr_def log_def)
```
```   201   also from c have "c powr log c K \<le> c powr real n" unfolding n_def
```
```   202     by (intro powr_mono, linarith, simp)
```
```   203   finally have "K \<le> c ^ n" using c by (simp add: powr_realpow)
```
```   204   also from c m have "c ^ n < c ^ m" by simp
```
```   205   also from c m have "c ^ m < root m (norm (f m)) ^ m" by (intro power_strict_mono) simp_all
```
```   206   also from m have "... = norm (f m)" by simp
```
```   207   finally show False using K(2)[of m]  by simp
```
```   208 qed
```
```   209
```
```   210
```
```   211 subsubsection \<open>Cauchy's condensation test\<close>
```
```   212
```
```   213 context
```
```   214 fixes f :: "nat \<Rightarrow> real"
```
```   215 begin
```
```   216
```
```   217 private lemma condensation_inequality:
```
```   218   assumes mono: "\<And>m n. 0 < m \<Longrightarrow> m \<le> n \<Longrightarrow> f n \<le> f m"
```
```   219   shows   "(\<Sum>k=1..<n. f k) \<ge> (\<Sum>k=1..<n. f (2 * 2 ^ natlog2 k))" (is "?thesis1")
```
```   220           "(\<Sum>k=1..<n. f k) \<le> (\<Sum>k=1..<n. f (2 ^ natlog2 k))" (is "?thesis2")
```
```   221   by (intro setsum_mono mono pow_natlog2_ge pow_natlog2_le, simp, simp)+
```
```   222
```
```   223 private lemma condensation_condense1: "(\<Sum>k=1..<2^n. f (2 ^ natlog2 k)) = (\<Sum>k<n. 2^k * f (2 ^ k))"
```
```   224 proof (induction n)
```
```   225   case (Suc n)
```
```   226   have "{1..<2^Suc n} = {1..<2^n} \<union> {2^n..<(2^Suc n :: nat)}" by auto
```
```   227   also have "(\<Sum>k\<in>\<dots>. f (2 ^ natlog2 k)) =
```
```   228                  (\<Sum>k<n. 2^k * f (2^k)) + (\<Sum>k = 2^n..<2^Suc n. f (2^natlog2 k))"
```
```   229     by (subst setsum.union_disjoint) (insert Suc, auto)
```
```   230   also have "natlog2 k = n" if "k \<in> {2^n..<2^Suc n}" for k using that by (intro natlog2_eqI) simp_all
```
```   231   hence "(\<Sum>k = 2^n..<2^Suc n. f (2^natlog2 k)) = (\<Sum>(_::nat) = 2^n..<2^Suc n. f (2^n))"
```
```   232     by (intro setsum.cong) simp_all
```
```   233   also have "\<dots> = 2^n * f (2^n)" by (simp add: of_nat_power)
```
```   234   finally show ?case by simp
```
```   235 qed simp
```
```   236
```
```   237 private lemma condensation_condense2: "(\<Sum>k=1..<2^n. f (2 * 2 ^ natlog2 k)) = (\<Sum>k<n. 2^k * f (2 ^ Suc k))"
```
```   238 proof (induction n)
```
```   239   case (Suc n)
```
```   240   have "{1..<2^Suc n} = {1..<2^n} \<union> {2^n..<(2^Suc n :: nat)}" by auto
```
```   241   also have "(\<Sum>k\<in>\<dots>. f (2 * 2 ^ natlog2 k)) =
```
```   242                  (\<Sum>k<n. 2^k * f (2^Suc k)) + (\<Sum>k = 2^n..<2^Suc n. f (2 * 2^natlog2 k))"
```
```   243     by (subst setsum.union_disjoint) (insert Suc, auto)
```
```   244   also have "natlog2 k = n" if "k \<in> {2^n..<2^Suc n}" for k using that by (intro natlog2_eqI) simp_all
```
```   245   hence "(\<Sum>k = 2^n..<2^Suc n. f (2*2^natlog2 k)) = (\<Sum>(_::nat) = 2^n..<2^Suc n. f (2^Suc n))"
```
```   246     by (intro setsum.cong) simp_all
```
```   247   also have "\<dots> = 2^n * f (2^Suc n)" by (simp add: of_nat_power)
```
```   248   finally show ?case by simp
```
```   249 qed simp
```
```   250
```
```   251 lemma condensation_test:
```
```   252   assumes mono: "\<And>m. 0 < m \<Longrightarrow> f (Suc m) \<le> f m"
```
```   253   assumes nonneg: "\<And>n. f n \<ge> 0"
```
```   254   shows "summable f \<longleftrightarrow> summable (\<lambda>n. 2^n * f (2^n))"
```
```   255 proof -
```
```   256   define f' where "f' n = (if n = 0 then 0 else f n)" for n
```
```   257   from mono have mono': "decseq (\<lambda>n. f (Suc n))" by (intro decseq_SucI) simp
```
```   258   hence mono': "f n \<le> f m" if "m \<le> n" "m > 0" for m n
```
```   259     using that decseqD[OF mono', of "m - 1" "n - 1"] by simp
```
```   260
```
```   261   have "(\<lambda>n. f (Suc n)) = (\<lambda>n. f' (Suc n))" by (intro ext) (simp add: f'_def)
```
```   262   hence "summable f \<longleftrightarrow> summable f'"
```
```   263     by (subst (1 2) summable_Suc_iff [symmetric]) (simp only:)
```
```   264   also have "\<dots> \<longleftrightarrow> convergent (\<lambda>n. \<Sum>k<n. f' k)" unfolding summable_iff_convergent ..
```
```   265   also have "monoseq (\<lambda>n. \<Sum>k<n. f' k)" unfolding f'_def
```
```   266     by (intro mono_SucI1) (auto intro!: mult_nonneg_nonneg nonneg)
```
```   267   hence "convergent (\<lambda>n. \<Sum>k<n. f' k) \<longleftrightarrow> Bseq (\<lambda>n. \<Sum>k<n. f' k)"
```
```   268     by (rule monoseq_imp_convergent_iff_Bseq)
```
```   269   also have "\<dots> \<longleftrightarrow> Bseq (\<lambda>n. \<Sum>k=1..<n. f' k)" unfolding One_nat_def
```
```   270     by (subst setsum_shift_lb_Suc0_0_upt) (simp_all add: f'_def atLeast0LessThan)
```
```   271   also have "\<dots> \<longleftrightarrow> Bseq (\<lambda>n. \<Sum>k=1..<n. f k)" unfolding f'_def by simp
```
```   272   also have "\<dots> \<longleftrightarrow> Bseq (\<lambda>n. \<Sum>k=1..<2^n. f k)"
```
```   273     by (rule nonneg_incseq_Bseq_subseq_iff[symmetric])
```
```   274        (auto intro!: setsum_nonneg incseq_SucI nonneg simp: subseq_def)
```
```   275   also have "\<dots> \<longleftrightarrow> Bseq (\<lambda>n. \<Sum>k<n. 2^k * f (2^k))"
```
```   276   proof (intro iffI)
```
```   277     assume A: "Bseq (\<lambda>n. \<Sum>k=1..<2^n. f k)"
```
```   278     have "eventually (\<lambda>n. norm (\<Sum>k<n. 2^k * f (2^Suc k)) \<le> norm (\<Sum>k=1..<2^n. f k)) sequentially"
```
```   279     proof (intro always_eventually allI)
```
```   280       fix n :: nat
```
```   281       have "norm (\<Sum>k<n. 2^k * f (2^Suc k)) = (\<Sum>k<n. 2^k * f (2^Suc k))" unfolding real_norm_def
```
```   282         by (intro abs_of_nonneg setsum_nonneg ballI mult_nonneg_nonneg nonneg) simp_all
```
```   283       also have "\<dots> \<le> (\<Sum>k=1..<2^n. f k)"
```
```   284         by (subst condensation_condense2 [symmetric]) (intro condensation_inequality mono')
```
```   285       also have "\<dots> = norm \<dots>" unfolding real_norm_def
```
```   286         by (intro abs_of_nonneg[symmetric] setsum_nonneg ballI mult_nonneg_nonneg nonneg)
```
```   287       finally show "norm (\<Sum>k<n. 2 ^ k * f (2 ^ Suc k)) \<le> norm (\<Sum>k=1..<2^n. f k)" .
```
```   288     qed
```
```   289     from this and A have "Bseq (\<lambda>n. \<Sum>k<n. 2^k * f (2^Suc k))" by (rule Bseq_eventually_mono)
```
```   290     from Bseq_mult[OF Bfun_const[of 2] this] have "Bseq (\<lambda>n. \<Sum>k<n. 2^Suc k * f (2^Suc k))"
```
```   291       by (simp add: setsum_right_distrib setsum_left_distrib mult_ac)
```
```   292     hence "Bseq (\<lambda>n. (\<Sum>k=Suc 0..<Suc n. 2^k * f (2^k)) + f 1)"
```
```   293       by (intro Bseq_add, subst setsum_shift_bounds_Suc_ivl) (simp add: atLeast0LessThan)
```
```   294     hence "Bseq (\<lambda>n. (\<Sum>k=0..<Suc n. 2^k * f (2^k)))"
```
```   295       by (subst setsum_head_upt_Suc) (simp_all add: add_ac)
```
```   296     thus "Bseq (\<lambda>n. (\<Sum>k<n. 2^k * f (2^k)))"
```
```   297       by (subst (asm) Bseq_Suc_iff) (simp add: atLeast0LessThan)
```
```   298   next
```
```   299     assume A: "Bseq (\<lambda>n. (\<Sum>k<n. 2^k * f (2^k)))"
```
```   300     have "eventually (\<lambda>n. norm (\<Sum>k=1..<2^n. f k) \<le> norm (\<Sum>k<n. 2^k * f (2^k))) sequentially"
```
```   301     proof (intro always_eventually allI)
```
```   302       fix n :: nat
```
```   303       have "norm (\<Sum>k=1..<2^n. f k) = (\<Sum>k=1..<2^n. f k)" unfolding real_norm_def
```
```   304         by (intro abs_of_nonneg setsum_nonneg ballI mult_nonneg_nonneg nonneg)
```
```   305       also have "\<dots> \<le> (\<Sum>k<n. 2^k * f (2^k))"
```
```   306         by (subst condensation_condense1 [symmetric]) (intro condensation_inequality mono')
```
```   307       also have "\<dots> = norm \<dots>" unfolding real_norm_def
```
```   308         by (intro abs_of_nonneg [symmetric] setsum_nonneg ballI mult_nonneg_nonneg nonneg) simp_all
```
```   309       finally show "norm (\<Sum>k=1..<2^n. f k) \<le> norm (\<Sum>k<n. 2^k * f (2^k))" .
```
```   310     qed
```
```   311     from this and A show "Bseq (\<lambda>n. \<Sum>k=1..<2^n. f k)" by (rule Bseq_eventually_mono)
```
```   312   qed
```
```   313   also have "monoseq (\<lambda>n. (\<Sum>k<n. 2^k * f (2^k)))"
```
```   314     by (intro mono_SucI1) (auto intro!: mult_nonneg_nonneg nonneg)
```
```   315   hence "Bseq (\<lambda>n. (\<Sum>k<n. 2^k * f (2^k))) \<longleftrightarrow> convergent (\<lambda>n. (\<Sum>k<n. 2^k * f (2^k)))"
```
```   316     by (rule monoseq_imp_convergent_iff_Bseq [symmetric])
```
```   317   also have "\<dots> \<longleftrightarrow> summable (\<lambda>k. 2^k * f (2^k))" by (simp only: summable_iff_convergent)
```
```   318   finally show ?thesis .
```
```   319 qed
```
```   320
```
```   321 end
```
```   322
```
```   323
```
```   324 subsubsection \<open>Summability of powers\<close>
```
```   325
```
```   326 lemma abs_summable_complex_powr_iff:
```
```   327     "summable (\<lambda>n. norm (exp (of_real (ln (of_nat n)) * s))) \<longleftrightarrow> Re s < -1"
```
```   328 proof (cases "Re s \<le> 0")
```
```   329   let ?l = "\<lambda>n. complex_of_real (ln (of_nat n))"
```
```   330   case False
```
```   331   with eventually_gt_at_top[of "0::nat"]
```
```   332     have "eventually (\<lambda>n. norm (1 :: real) \<le> norm (exp (?l n * s))) sequentially"
```
```   333     by (auto intro!: ge_one_powr_ge_zero elim!: eventually_mono)
```
```   334   from summable_comparison_test_ev[OF this] False show ?thesis by (auto simp: summable_const_iff)
```
```   335 next
```
```   336   let ?l = "\<lambda>n. complex_of_real (ln (of_nat n))"
```
```   337   case True
```
```   338   hence "summable (\<lambda>n. norm (exp (?l n * s))) \<longleftrightarrow> summable (\<lambda>n. 2^n * norm (exp (?l (2^n) * s)))"
```
```   339     by (intro condensation_test) (auto intro!: mult_right_mono_neg)
```
```   340   also have "(\<lambda>n. 2^n * norm (exp (?l (2^n) * s))) = (\<lambda>n. (2 powr (Re s + 1)) ^ n)"
```
```   341   proof
```
```   342     fix n :: nat
```
```   343     have "2^n * norm (exp (?l (2^n) * s)) = exp (real n * ln 2) * exp (real n * ln 2 * Re s)"
```
```   344       using True by (subst exp_of_nat_mult) (simp add: ln_realpow algebra_simps)
```
```   345     also have "\<dots> = exp (real n * (ln 2 * (Re s + 1)))"
```
```   346       by (simp add: algebra_simps exp_add)
```
```   347     also have "\<dots> = exp (ln 2 * (Re s + 1)) ^ n" by (subst exp_of_nat_mult) simp
```
```   348     also have "exp (ln 2 * (Re s + 1)) = 2 powr (Re s + 1)" by (simp add: powr_def)
```
```   349     finally show "2^n * norm (exp (?l (2^n) * s)) = (2 powr (Re s + 1)) ^ n" .
```
```   350   qed
```
```   351   also have "summable \<dots> \<longleftrightarrow> 2 powr (Re s + 1) < 2 powr 0"
```
```   352     by (subst summable_geometric_iff) simp
```
```   353   also have "\<dots> \<longleftrightarrow> Re s < -1" by (subst powr_less_cancel_iff) (simp, linarith)
```
```   354   finally show ?thesis .
```
```   355 qed
```
```   356
```
```   357 lemma summable_complex_powr_iff:
```
```   358   assumes "Re s < -1"
```
```   359   shows   "summable (\<lambda>n. exp (of_real (ln (of_nat n)) * s))"
```
```   360   by (rule summable_norm_cancel, subst abs_summable_complex_powr_iff) fact
```
```   361
```
```   362 lemma summable_real_powr_iff: "summable (\<lambda>n. of_nat n powr s :: real) \<longleftrightarrow> s < -1"
```
```   363 proof -
```
```   364   from eventually_gt_at_top[of "0::nat"]
```
```   365     have "summable (\<lambda>n. of_nat n powr s) \<longleftrightarrow> summable (\<lambda>n. exp (ln (of_nat n) * s))"
```
```   366     by (intro summable_cong) (auto elim!: eventually_mono simp: powr_def)
```
```   367   also have "\<dots> \<longleftrightarrow> s < -1" using abs_summable_complex_powr_iff[of "of_real s"] by simp
```
```   368   finally show ?thesis .
```
```   369 qed
```
```   370
```
```   371 lemma inverse_power_summable:
```
```   372   assumes s: "s \<ge> 2"
```
```   373   shows "summable (\<lambda>n. inverse (of_nat n ^ s :: 'a :: {real_normed_div_algebra,banach}))"
```
```   374 proof (rule summable_norm_cancel, subst summable_cong)
```
```   375   from eventually_gt_at_top[of "0::nat"]
```
```   376     show "eventually (\<lambda>n. norm (inverse (of_nat n ^ s:: 'a)) = real_of_nat n powr (-real s)) at_top"
```
```   377     by eventually_elim (simp add: norm_inverse norm_power powr_minus powr_realpow)
```
```   378 qed (insert s summable_real_powr_iff[of "-s"], simp_all)
```
```   379
```
```   380 lemma not_summable_harmonic: "\<not>summable (\<lambda>n. inverse (of_nat n) :: 'a :: real_normed_field)"
```
```   381 proof
```
```   382   assume "summable (\<lambda>n. inverse (of_nat n) :: 'a)"
```
```   383   hence "convergent (\<lambda>n. norm (of_real (\<Sum>k<n. inverse (of_nat k)) :: 'a))"
```
```   384     by (simp add: summable_iff_convergent convergent_norm)
```
```   385   hence "convergent (\<lambda>n. abs (\<Sum>k<n. inverse (of_nat k)) :: real)" by (simp only: norm_of_real)
```
```   386   also have "(\<lambda>n. abs (\<Sum>k<n. inverse (of_nat k)) :: real) = (\<lambda>n. \<Sum>k<n. inverse (of_nat k))"
```
```   387     by (intro ext abs_of_nonneg setsum_nonneg) auto
```
```   388   also have "convergent \<dots> \<longleftrightarrow> summable (\<lambda>k. inverse (of_nat k) :: real)"
```
```   389     by (simp add: summable_iff_convergent)
```
```   390   finally show False using summable_real_powr_iff[of "-1"] by (simp add: powr_minus)
```
```   391 qed
```
```   392
```
```   393
```
```   394 subsubsection \<open>Kummer's test\<close>
```
```   395
```
```   396 lemma kummers_test_convergence:
```
```   397   fixes f p :: "nat \<Rightarrow> real"
```
```   398   assumes pos_f: "eventually (\<lambda>n. f n > 0) sequentially"
```
```   399   assumes nonneg_p: "eventually (\<lambda>n. p n \<ge> 0) sequentially"
```
```   400   defines "l \<equiv> liminf (\<lambda>n. ereal (p n * f n / f (Suc n) - p (Suc n)))"
```
```   401   assumes l: "l > 0"
```
```   402   shows   "summable f"
```
```   403   unfolding summable_iff_convergent'
```
```   404 proof -
```
```   405   define r where "r = (if l = \<infinity> then 1 else real_of_ereal l / 2)"
```
```   406   from l have "r > 0 \<and> of_real r < l" by (cases l) (simp_all add: r_def)
```
```   407   hence r: "r > 0" "of_real r < l" by simp_all
```
```   408   hence "eventually (\<lambda>n. p n * f n / f (Suc n) - p (Suc n) > r) sequentially"
```
```   409     unfolding l_def by (force dest: less_LiminfD)
```
```   410   moreover from pos_f have "eventually (\<lambda>n. f (Suc n) > 0) sequentially"
```
```   411     by (subst eventually_sequentially_Suc)
```
```   412   ultimately have "eventually (\<lambda>n. p n * f n - p (Suc n) * f (Suc n) > r * f (Suc n)) sequentially"
```
```   413     by eventually_elim (simp add: field_simps)
```
```   414   from eventually_conj[OF pos_f eventually_conj[OF nonneg_p this]]
```
```   415     obtain m where m: "\<And>n. n \<ge> m \<Longrightarrow> f n > 0" "\<And>n. n \<ge> m \<Longrightarrow> p n \<ge> 0"
```
```   416         "\<And>n. n \<ge> m \<Longrightarrow> p n * f n - p (Suc n) * f (Suc n) > r * f (Suc n)"
```
```   417     unfolding eventually_at_top_linorder by blast
```
```   418
```
```   419   let ?c = "(norm (\<Sum>k\<le>m. r * f k) + p m * f m) / r"
```
```   420   have "Bseq (\<lambda>n. (\<Sum>k\<le>n + Suc m. f k))"
```
```   421   proof (rule BseqI')
```
```   422     fix k :: nat
```
```   423     define n where "n = k + Suc m"
```
```   424     have n: "n > m" by (simp add: n_def)
```
```   425
```
```   426     from r have "r * norm (\<Sum>k\<le>n. f k) = norm (\<Sum>k\<le>n. r * f k)"
```
```   427       by (simp add: setsum_right_distrib[symmetric] abs_mult)
```
```   428     also from n have "{..n} = {..m} \<union> {Suc m..n}" by auto
```
```   429     hence "(\<Sum>k\<le>n. r * f k) = (\<Sum>k\<in>{..m} \<union> {Suc m..n}. r * f k)" by (simp only:)
```
```   430     also have "\<dots> = (\<Sum>k\<le>m. r * f k) + (\<Sum>k=Suc m..n. r * f k)"
```
```   431       by (subst setsum.union_disjoint) auto
```
```   432     also have "norm \<dots> \<le> norm (\<Sum>k\<le>m. r * f k) + norm (\<Sum>k=Suc m..n. r * f k)"
```
```   433       by (rule norm_triangle_ineq)
```
```   434     also from r less_imp_le[OF m(1)] have "(\<Sum>k=Suc m..n. r * f k) \<ge> 0"
```
```   435       by (intro setsum_nonneg) auto
```
```   436     hence "norm (\<Sum>k=Suc m..n. r * f k) = (\<Sum>k=Suc m..n. r * f k)" by simp
```
```   437     also have "(\<Sum>k=Suc m..n. r * f k) = (\<Sum>k=m..<n. r * f (Suc k))"
```
```   438      by (subst setsum_shift_bounds_Suc_ivl [symmetric])
```
```   439           (simp only: atLeastLessThanSuc_atLeastAtMost)
```
```   440     also from m have "\<dots> \<le> (\<Sum>k=m..<n. p k * f k - p (Suc k) * f (Suc k))"
```
```   441       by (intro setsum_mono[OF less_imp_le]) simp_all
```
```   442     also have "\<dots> = -(\<Sum>k=m..<n. p (Suc k) * f (Suc k) - p k * f k)"
```
```   443       by (simp add: setsum_negf [symmetric] algebra_simps)
```
```   444     also from n have "\<dots> = p m * f m - p n * f n"
```
```   445       by (cases n, simp, simp only: atLeastLessThanSuc_atLeastAtMost, subst setsum_Suc_diff) simp_all
```
```   446     also from less_imp_le[OF m(1)] m(2) n have "\<dots> \<le> p m * f m" by simp
```
```   447     finally show "norm (\<Sum>k\<le>n. f k) \<le> (norm (\<Sum>k\<le>m. r * f k) + p m * f m) / r" using r
```
```   448       by (subst pos_le_divide_eq[OF r(1)]) (simp only: mult_ac)
```
```   449   qed
```
```   450   moreover have "(\<Sum>k\<le>n. f k) \<le> (\<Sum>k\<le>n'. f k)" if "Suc m \<le> n" "n \<le> n'" for n n'
```
```   451     using less_imp_le[OF m(1)] that by (intro setsum_mono2) auto
```
```   452   ultimately show "convergent (\<lambda>n. \<Sum>k\<le>n. f k)" by (rule Bseq_monoseq_convergent'_inc)
```
```   453 qed
```
```   454
```
```   455
```
```   456 lemma kummers_test_divergence:
```
```   457   fixes f p :: "nat \<Rightarrow> real"
```
```   458   assumes pos_f: "eventually (\<lambda>n. f n > 0) sequentially"
```
```   459   assumes pos_p: "eventually (\<lambda>n. p n > 0) sequentially"
```
```   460   assumes divergent_p: "\<not>summable (\<lambda>n. inverse (p n))"
```
```   461   defines "l \<equiv> limsup (\<lambda>n. ereal (p n * f n / f (Suc n) - p (Suc n)))"
```
```   462   assumes l: "l < 0"
```
```   463   shows   "\<not>summable f"
```
```   464 proof
```
```   465   assume "summable f"
```
```   466   from eventually_conj[OF pos_f eventually_conj[OF pos_p Limsup_lessD[OF l[unfolded l_def]]]]
```
```   467     obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> p n > 0" "\<And>n. n \<ge> N \<Longrightarrow> f n > 0"
```
```   468                       "\<And>n. n \<ge> N \<Longrightarrow> p n * f n / f (Suc n) - p (Suc n) < 0"
```
```   469     by (auto simp: eventually_at_top_linorder)
```
```   470   hence A: "p n * f n < p (Suc n) * f (Suc n)" if "n \<ge> N" for n using that N[of n] N[of "Suc n"]
```
```   471     by (simp add: field_simps)
```
```   472   have "p n * f n \<ge> p N * f N" if "n \<ge> N" for n using that and A
```
```   473       by (induction n rule: dec_induct) (auto intro!: less_imp_le elim!: order.trans)
```
```   474   from eventually_ge_at_top[of N] N this
```
```   475     have "eventually (\<lambda>n. norm (p N * f N * inverse (p n)) \<le> f n) sequentially"
```
```   476     by (auto elim!: eventually_mono simp: field_simps abs_of_pos)
```
```   477   from this and \<open>summable f\<close> have "summable (\<lambda>n. p N * f N * inverse (p n))"
```
```   478     by (rule summable_comparison_test_ev)
```
```   479   from summable_mult[OF this, of "inverse (p N * f N)"] N[OF le_refl]
```
```   480     have "summable (\<lambda>n. inverse (p n))" by (simp add: divide_simps)
```
```   481   with divergent_p show False by contradiction
```
```   482 qed
```
```   483
```
```   484
```
```   485 subsubsection \<open>Ratio test\<close>
```
```   486
```
```   487 lemma ratio_test_convergence:
```
```   488   fixes f :: "nat \<Rightarrow> real"
```
```   489   assumes pos_f: "eventually (\<lambda>n. f n > 0) sequentially"
```
```   490   defines "l \<equiv> liminf (\<lambda>n. ereal (f n / f (Suc n)))"
```
```   491   assumes l: "l > 1"
```
```   492   shows   "summable f"
```
```   493 proof (rule kummers_test_convergence[OF pos_f])
```
```   494   note l
```
```   495   also have "l = liminf (\<lambda>n. ereal (f n / f (Suc n) - 1)) + 1"
```
```   496     by (subst Liminf_add_ereal_right[symmetric]) (simp_all add: minus_ereal_def l_def one_ereal_def)
```
```   497   finally show "liminf (\<lambda>n. ereal (1 * f n / f (Suc n) - 1)) > 0"
```
```   498     by (cases "liminf (\<lambda>n. ereal (1 * f n / f (Suc n) - 1))") simp_all
```
```   499 qed simp
```
```   500
```
```   501 lemma ratio_test_divergence:
```
```   502   fixes f :: "nat \<Rightarrow> real"
```
```   503   assumes pos_f: "eventually (\<lambda>n. f n > 0) sequentially"
```
```   504   defines "l \<equiv> limsup (\<lambda>n. ereal (f n / f (Suc n)))"
```
```   505   assumes l: "l < 1"
```
```   506   shows   "\<not>summable f"
```
```   507 proof (rule kummers_test_divergence[OF pos_f])
```
```   508   have "limsup (\<lambda>n. ereal (f n / f (Suc n) - 1)) + 1 = l"
```
```   509     by (subst Limsup_add_ereal_right[symmetric]) (simp_all add: minus_ereal_def l_def one_ereal_def)
```
```   510   also note l
```
```   511   finally show "limsup (\<lambda>n. ereal (1 * f n / f (Suc n) - 1)) < 0"
```
```   512     by (cases "limsup (\<lambda>n. ereal (1 * f n / f (Suc n) - 1))") simp_all
```
```   513 qed (simp_all add: summable_const_iff)
```
```   514
```
```   515
```
```   516 subsubsection \<open>Raabe's test\<close>
```
```   517
```
```   518 lemma raabes_test_convergence:
```
```   519 fixes f :: "nat \<Rightarrow> real"
```
```   520   assumes pos: "eventually (\<lambda>n. f n > 0) sequentially"
```
```   521   defines "l \<equiv> liminf (\<lambda>n. ereal (of_nat n * (f n / f (Suc n) - 1)))"
```
```   522   assumes l: "l > 1"
```
```   523   shows   "summable f"
```
```   524 proof (rule kummers_test_convergence)
```
```   525   let ?l' = "liminf (\<lambda>n. ereal (of_nat n * f n / f (Suc n) - of_nat (Suc n)))"
```
```   526   have "1 < l" by fact
```
```   527   also have "l = liminf (\<lambda>n. ereal (of_nat n * f n / f (Suc n) - of_nat (Suc n)) + 1)"
```
```   528     by (simp add: l_def algebra_simps)
```
```   529   also have "\<dots> = ?l' + 1" by (subst Liminf_add_ereal_right) simp_all
```
```   530   finally show "?l' > 0" by (cases ?l') (simp_all add: algebra_simps)
```
```   531 qed (simp_all add: pos)
```
```   532
```
```   533 lemma raabes_test_divergence:
```
```   534 fixes f :: "nat \<Rightarrow> real"
```
```   535   assumes pos: "eventually (\<lambda>n. f n > 0) sequentially"
```
```   536   defines "l \<equiv> limsup (\<lambda>n. ereal (of_nat n * (f n / f (Suc n) - 1)))"
```
```   537   assumes l: "l < 1"
```
```   538   shows   "\<not>summable f"
```
```   539 proof (rule kummers_test_divergence)
```
```   540   let ?l' = "limsup (\<lambda>n. ereal (of_nat n * f n / f (Suc n) - of_nat (Suc n)))"
```
```   541   note l
```
```   542   also have "l = limsup (\<lambda>n. ereal (of_nat n * f n / f (Suc n) - of_nat (Suc n)) + 1)"
```
```   543     by (simp add: l_def algebra_simps)
```
```   544   also have "\<dots> = ?l' + 1" by (subst Limsup_add_ereal_right) simp_all
```
```   545   finally show "?l' < 0" by (cases ?l') (simp_all add: algebra_simps)
```
```   546 qed (insert pos eventually_gt_at_top[of "0::nat"] not_summable_harmonic, simp_all)
```
```   547
```
```   548
```
```   549
```
```   550 subsection \<open>Radius of convergence\<close>
```
```   551
```
```   552 text \<open>
```
```   553   The radius of convergence of a power series. This value always exists, ranges from
```
```   554   @{term "0::ereal"} to @{term "\<infinity>::ereal"}, and the power series is guaranteed to converge for
```
```   555   all inputs with a norm that is smaller than that radius and to diverge for all inputs with a
```
```   556   norm that is greater.
```
```   557 \<close>
```
```   558 definition conv_radius :: "(nat \<Rightarrow> 'a :: banach) \<Rightarrow> ereal" where
```
```   559   "conv_radius f = inverse (limsup (\<lambda>n. ereal (root n (norm (f n)))))"
```
```   560
```
```   561 lemma conv_radius_nonneg: "conv_radius f \<ge> 0"
```
```   562 proof -
```
```   563   have "0 = limsup (\<lambda>n. 0)" by (subst Limsup_const) simp_all
```
```   564   also have "\<dots> \<le> limsup (\<lambda>n. ereal (root n (norm (f n))))"
```
```   565     by (intro Limsup_mono) (simp_all add: real_root_ge_zero)
```
```   566   finally show ?thesis
```
```   567     unfolding conv_radius_def by (auto simp: ereal_inverse_nonneg_iff)
```
```   568 qed
```
```   569
```
```   570 lemma conv_radius_zero [simp]: "conv_radius (\<lambda>_. 0) = \<infinity>"
```
```   571   by (auto simp: conv_radius_def zero_ereal_def [symmetric] Limsup_const)
```
```   572
```
```   573 lemma conv_radius_cong:
```
```   574   assumes "eventually (\<lambda>x. f x = g x) sequentially"
```
```   575   shows   "conv_radius f = conv_radius g"
```
```   576 proof -
```
```   577   have "eventually (\<lambda>n. ereal (root n (norm (f n))) = ereal (root n (norm (g n)))) sequentially"
```
```   578     using assms by eventually_elim simp
```
```   579   from Limsup_eq[OF this] show ?thesis unfolding conv_radius_def by simp
```
```   580 qed
```
```   581
```
```   582 lemma conv_radius_altdef:
```
```   583   "conv_radius f = liminf (\<lambda>n. inverse (ereal (root n (norm (f n)))))"
```
```   584   by (subst Liminf_inverse_ereal) (simp_all add: real_root_ge_zero conv_radius_def)
```
```   585
```
```   586 lemma abs_summable_in_conv_radius:
```
```   587   fixes f :: "nat \<Rightarrow> 'a :: {banach, real_normed_div_algebra}"
```
```   588   assumes "ereal (norm z) < conv_radius f"
```
```   589   shows   "summable (\<lambda>n. norm (f n * z ^ n))"
```
```   590 proof (rule root_test_convergence')
```
```   591   define l where "l = limsup (\<lambda>n. ereal (root n (norm (f n))))"
```
```   592   have "0 = limsup (\<lambda>n. 0)" by (simp add: Limsup_const)
```
```   593   also have "... \<le> l" unfolding l_def by (intro Limsup_mono) (simp_all add: real_root_ge_zero)
```
```   594   finally have l_nonneg: "l \<ge> 0" .
```
```   595
```
```   596   have "limsup (\<lambda>n. root n (norm (f n * z^n))) = l * ereal (norm z)" unfolding l_def
```
```   597     by (rule limsup_root_powser)
```
```   598   also from l_nonneg consider "l = 0" | "l = \<infinity>" | "\<exists>l'. l = ereal l' \<and> l' > 0"
```
```   599     by (cases "l") (auto simp: less_le)
```
```   600   hence "l * ereal (norm z) < 1"
```
```   601   proof cases
```
```   602     assume "l = \<infinity>"
```
```   603     hence "conv_radius f = 0" unfolding conv_radius_def l_def by simp
```
```   604     with assms show ?thesis by simp
```
```   605   next
```
```   606     assume "\<exists>l'. l = ereal l' \<and> l' > 0"
```
```   607     then guess l' by (elim exE conjE) note l' = this
```
```   608     hence "l \<noteq> \<infinity>" by auto
```
```   609     have "l * ereal (norm z) < l * conv_radius f"
```
```   610       by (intro ereal_mult_strict_left_mono) (simp_all add: l' assms)
```
```   611     also have "conv_radius f = inverse l" by (simp add: conv_radius_def l_def)
```
```   612     also from l' have "l * inverse l = 1" by simp
```
```   613     finally show ?thesis .
```
```   614   qed simp_all
```
```   615   finally show "limsup (\<lambda>n. ereal (root n (norm (norm (f n * z ^ n))))) < 1" by simp
```
```   616 qed
```
```   617
```
```   618 lemma summable_in_conv_radius:
```
```   619   fixes f :: "nat \<Rightarrow> 'a :: {banach, real_normed_div_algebra}"
```
```   620   assumes "ereal (norm z) < conv_radius f"
```
```   621   shows   "summable (\<lambda>n. f n * z ^ n)"
```
```   622   by (rule summable_norm_cancel, rule abs_summable_in_conv_radius) fact+
```
```   623
```
```   624 lemma not_summable_outside_conv_radius:
```
```   625   fixes f :: "nat \<Rightarrow> 'a :: {banach, real_normed_div_algebra}"
```
```   626   assumes "ereal (norm z) > conv_radius f"
```
```   627   shows   "\<not>summable (\<lambda>n. f n * z ^ n)"
```
```   628 proof (rule root_test_divergence)
```
```   629   define l where "l = limsup (\<lambda>n. ereal (root n (norm (f n))))"
```
```   630   have "0 = limsup (\<lambda>n. 0)" by (simp add: Limsup_const)
```
```   631   also have "... \<le> l" unfolding l_def by (intro Limsup_mono) (simp_all add: real_root_ge_zero)
```
```   632   finally have l_nonneg: "l \<ge> 0" .
```
```   633   from assms have l_nz: "l \<noteq> 0" unfolding conv_radius_def l_def by auto
```
```   634
```
```   635   have "limsup (\<lambda>n. ereal (root n (norm (f n * z^n)))) = l * ereal (norm z)"
```
```   636     unfolding l_def by (rule limsup_root_powser)
```
```   637   also have "... > 1"
```
```   638   proof (cases l)
```
```   639     assume "l = \<infinity>"
```
```   640     with assms conv_radius_nonneg[of f] show ?thesis
```
```   641       by (auto simp: zero_ereal_def[symmetric])
```
```   642   next
```
```   643     fix l' assume l': "l = ereal l'"
```
```   644     from l_nonneg l_nz have "1 = l * inverse l" by (auto simp: l' field_simps)
```
```   645     also from l_nz have "inverse l = conv_radius f"
```
```   646       unfolding l_def conv_radius_def by auto
```
```   647     also from l' l_nz l_nonneg assms have "l * \<dots> < l * ereal (norm z)"
```
```   648       by (intro ereal_mult_strict_left_mono) (auto simp: l')
```
```   649     finally show ?thesis .
```
```   650   qed (insert l_nonneg, simp_all)
```
```   651   finally show "limsup (\<lambda>n. ereal (root n (norm (f n * z^n)))) > 1" .
```
```   652 qed
```
```   653
```
```   654
```
```   655 lemma conv_radius_geI:
```
```   656   assumes "summable (\<lambda>n. f n * z ^ n :: 'a :: {banach, real_normed_div_algebra})"
```
```   657   shows   "conv_radius f \<ge> norm z"
```
```   658   using not_summable_outside_conv_radius[of f z] assms by (force simp: not_le[symmetric])
```
```   659
```
```   660 lemma conv_radius_leI:
```
```   661   assumes "\<not>summable (\<lambda>n. norm (f n * z ^ n :: 'a :: {banach, real_normed_div_algebra}))"
```
```   662   shows   "conv_radius f \<le> norm z"
```
```   663   using abs_summable_in_conv_radius[of z f] assms by (force simp: not_le[symmetric])
```
```   664
```
```   665 lemma conv_radius_leI':
```
```   666   assumes "\<not>summable (\<lambda>n. f n * z ^ n :: 'a :: {banach, real_normed_div_algebra})"
```
```   667   shows   "conv_radius f \<le> norm z"
```
```   668   using summable_in_conv_radius[of z f] assms by (force simp: not_le[symmetric])
```
```   669
```
```   670 lemma conv_radius_geI_ex:
```
```   671   fixes f :: "nat \<Rightarrow> 'a :: {banach, real_normed_div_algebra}"
```
```   672   assumes "\<And>r. 0 < r \<Longrightarrow> ereal r < R \<Longrightarrow> \<exists>z. norm z = r \<and> summable (\<lambda>n. f n * z^n)"
```
```   673   shows   "conv_radius f \<ge> R"
```
```   674 proof (rule linorder_cases[of "conv_radius f" R])
```
```   675   assume R: "conv_radius f < R"
```
```   676   with conv_radius_nonneg[of f] obtain conv_radius'
```
```   677     where [simp]: "conv_radius f = ereal conv_radius'"
```
```   678     by (cases "conv_radius f") simp_all
```
```   679   define r where "r = (if R = \<infinity> then conv_radius' + 1 else (real_of_ereal R + conv_radius') / 2)"
```
```   680   from R conv_radius_nonneg[of f] have "0 < r \<and> ereal r < R \<and> ereal r > conv_radius f"
```
```   681     unfolding r_def by (cases R) (auto simp: r_def field_simps)
```
```   682   with assms(1)[of r] obtain z where "norm z > conv_radius f" "summable (\<lambda>n. f n * z^n)" by auto
```
```   683   with not_summable_outside_conv_radius[of f z] show ?thesis by simp
```
```   684 qed simp_all
```
```   685
```
```   686 lemma conv_radius_geI_ex':
```
```   687   fixes f :: "nat \<Rightarrow> 'a :: {banach, real_normed_div_algebra}"
```
```   688   assumes "\<And>r. 0 < r \<Longrightarrow> ereal r < R \<Longrightarrow> summable (\<lambda>n. f n * of_real r^n)"
```
```   689   shows   "conv_radius f \<ge> R"
```
```   690 proof (rule conv_radius_geI_ex)
```
```   691   fix r assume "0 < r" "ereal r < R"
```
```   692   with assms[of r] show "\<exists>z. norm z = r \<and> summable (\<lambda>n. f n * z ^ n)"
```
```   693     by (intro exI[of _ "of_real r :: 'a"]) auto
```
```   694 qed
```
```   695
```
```   696 lemma conv_radius_leI_ex:
```
```   697   fixes f :: "nat \<Rightarrow> 'a :: {banach, real_normed_div_algebra}"
```
```   698   assumes "R \<ge> 0"
```
```   699   assumes "\<And>r. 0 < r \<Longrightarrow> ereal r > R \<Longrightarrow> \<exists>z. norm z = r \<and> \<not>summable (\<lambda>n. norm (f n * z^n))"
```
```   700   shows   "conv_radius f \<le> R"
```
```   701 proof (rule linorder_cases[of "conv_radius f" R])
```
```   702   assume R: "conv_radius f > R"
```
```   703   from R assms(1) obtain R' where R': "R = ereal R'" by (cases R) simp_all
```
```   704   define r where
```
```   705     "r = (if conv_radius f = \<infinity> then R' + 1 else (R' + real_of_ereal (conv_radius f)) / 2)"
```
```   706   from R conv_radius_nonneg[of f] have "r > R \<and> r < conv_radius f" unfolding r_def
```
```   707     by (cases "conv_radius f") (auto simp: r_def field_simps R')
```
```   708   with assms(1) assms(2)[of r] R'
```
```   709     obtain z where "norm z < conv_radius f" "\<not>summable (\<lambda>n. norm (f n * z^n))" by auto
```
```   710   with abs_summable_in_conv_radius[of z f] show ?thesis by auto
```
```   711 qed simp_all
```
```   712
```
```   713 lemma conv_radius_leI_ex':
```
```   714   fixes f :: "nat \<Rightarrow> 'a :: {banach, real_normed_div_algebra}"
```
```   715   assumes "R \<ge> 0"
```
```   716   assumes "\<And>r. 0 < r \<Longrightarrow> ereal r > R \<Longrightarrow> \<not>summable (\<lambda>n. f n * of_real r^n)"
```
```   717   shows   "conv_radius f \<le> R"
```
```   718 proof (rule conv_radius_leI_ex)
```
```   719   fix r assume "0 < r" "ereal r > R"
```
```   720   with assms(2)[of r] show "\<exists>z. norm z = r \<and> \<not>summable (\<lambda>n. norm (f n * z ^ n))"
```
```   721     by (intro exI[of _ "of_real r :: 'a"]) (auto dest: summable_norm_cancel)
```
```   722 qed fact+
```
```   723
```
```   724 lemma conv_radius_eqI:
```
```   725   fixes f :: "nat \<Rightarrow> 'a :: {banach, real_normed_div_algebra}"
```
```   726   assumes "R \<ge> 0"
```
```   727   assumes "\<And>r. 0 < r \<Longrightarrow> ereal r < R \<Longrightarrow> \<exists>z. norm z = r \<and> summable (\<lambda>n. f n * z^n)"
```
```   728   assumes "\<And>r. 0 < r \<Longrightarrow> ereal r > R \<Longrightarrow> \<exists>z. norm z = r \<and> \<not>summable (\<lambda>n. norm (f n * z^n))"
```
```   729   shows   "conv_radius f = R"
```
```   730   by (intro antisym conv_radius_geI_ex conv_radius_leI_ex assms)
```
```   731
```
```   732 lemma conv_radius_eqI':
```
```   733   fixes f :: "nat \<Rightarrow> 'a :: {banach, real_normed_div_algebra}"
```
```   734   assumes "R \<ge> 0"
```
```   735   assumes "\<And>r. 0 < r \<Longrightarrow> ereal r < R \<Longrightarrow> summable (\<lambda>n. f n * (of_real r)^n)"
```
```   736   assumes "\<And>r. 0 < r \<Longrightarrow> ereal r > R \<Longrightarrow> \<not>summable (\<lambda>n. norm (f n * (of_real r)^n))"
```
```   737   shows   "conv_radius f = R"
```
```   738 proof (intro conv_radius_eqI[OF assms(1)])
```
```   739   fix r assume "0 < r" "ereal r < R" with assms(2)[OF this]
```
```   740     show "\<exists>z. norm z = r \<and> summable (\<lambda>n. f n * z ^ n)" by force
```
```   741 next
```
```   742   fix r assume "0 < r" "ereal r > R" with assms(3)[OF this]
```
```   743     show "\<exists>z. norm z = r \<and> \<not>summable (\<lambda>n. norm (f n * z ^ n))" by force
```
```   744 qed
```
```   745
```
```   746 lemma conv_radius_zeroI:
```
```   747   fixes f :: "nat \<Rightarrow> 'a :: {banach,real_normed_div_algebra}"
```
```   748   assumes "\<And>z. z \<noteq> 0 \<Longrightarrow> \<not>summable (\<lambda>n. f n * z^n)"
```
```   749   shows   "conv_radius f = 0"
```
```   750 proof (rule ccontr)
```
```   751   assume "conv_radius f \<noteq> 0"
```
```   752   with conv_radius_nonneg[of f] have pos: "conv_radius f > 0" by simp
```
```   753   define r where "r = (if conv_radius f = \<infinity> then 1 else real_of_ereal (conv_radius f) / 2)"
```
```   754   from pos have r: "ereal r > 0 \<and> ereal r < conv_radius f"
```
```   755     by (cases "conv_radius f") (simp_all add: r_def)
```
```   756   hence "summable (\<lambda>n. f n * of_real r ^ n)" by (intro summable_in_conv_radius) simp
```
```   757   moreover from r and assms[of "of_real r"] have "\<not>summable (\<lambda>n. f n * of_real r ^ n)" by simp
```
```   758   ultimately show False by contradiction
```
```   759 qed
```
```   760
```
```   761 lemma conv_radius_inftyI':
```
```   762   fixes f :: "nat \<Rightarrow> 'a :: {banach,real_normed_div_algebra}"
```
```   763   assumes "\<And>r. r > c \<Longrightarrow> \<exists>z. norm z = r \<and> summable (\<lambda>n. f n * z^n)"
```
```   764   shows   "conv_radius f = \<infinity>"
```
```   765 proof -
```
```   766   {
```
```   767     fix r :: real
```
```   768     have "max r (c + 1) > c" by (auto simp: max_def)
```
```   769     from assms[OF this] obtain z where "norm z = max r (c + 1)" "summable (\<lambda>n. f n * z^n)" by blast
```
```   770     from conv_radius_geI[OF this(2)] this(1) have "conv_radius f \<ge> r" by simp
```
```   771   }
```
```   772   from this[of "real_of_ereal (conv_radius f + 1)"] show "conv_radius f = \<infinity>"
```
```   773     by (cases "conv_radius f") simp_all
```
```   774 qed
```
```   775
```
```   776 lemma conv_radius_inftyI:
```
```   777   fixes f :: "nat \<Rightarrow> 'a :: {banach,real_normed_div_algebra}"
```
```   778   assumes "\<And>r. \<exists>z. norm z = r \<and> summable (\<lambda>n. f n * z^n)"
```
```   779   shows   "conv_radius f = \<infinity>"
```
```   780   using assms by (rule conv_radius_inftyI')
```
```   781
```
```   782 lemma conv_radius_inftyI'':
```
```   783   fixes f :: "nat \<Rightarrow> 'a :: {banach,real_normed_div_algebra}"
```
```   784   assumes "\<And>z. summable (\<lambda>n. f n * z^n)"
```
```   785   shows   "conv_radius f = \<infinity>"
```
```   786 proof (rule conv_radius_inftyI')
```
```   787   fix r :: real assume "r > 0"
```
```   788   with assms show "\<exists>z. norm z = r \<and> summable (\<lambda>n. f n * z^n)"
```
```   789     by (intro exI[of _ "of_real r"]) simp
```
```   790 qed
```
```   791
```
```   792 lemma conv_radius_ratio_limit_ereal:
```
```   793   fixes f :: "nat \<Rightarrow> 'a :: {banach,real_normed_div_algebra}"
```
```   794   assumes nz:  "eventually (\<lambda>n. f n \<noteq> 0) sequentially"
```
```   795   assumes lim: "(\<lambda>n. ereal (norm (f n) / norm (f (Suc n)))) \<longlonglongrightarrow> c"
```
```   796   shows   "conv_radius f = c"
```
```   797 proof (rule conv_radius_eqI')
```
```   798   show "c \<ge> 0" by (intro Lim_bounded2_ereal[OF lim]) simp_all
```
```   799 next
```
```   800   fix r assume r: "0 < r" "ereal r < c"
```
```   801   let ?l = "liminf (\<lambda>n. ereal (norm (f n * of_real r ^ n) / norm (f (Suc n) * of_real r ^ Suc n)))"
```
```   802   have "?l = liminf (\<lambda>n. ereal (norm (f n) / (norm (f (Suc n)))) * ereal (inverse r))"
```
```   803     using r by (simp add: norm_mult norm_power divide_simps)
```
```   804   also from r have "\<dots> = liminf (\<lambda>n. ereal (norm (f n) / (norm (f (Suc n))))) * ereal (inverse r)"
```
```   805     by (intro Liminf_ereal_mult_right) simp_all
```
```   806   also have "liminf (\<lambda>n. ereal (norm (f n) / (norm (f (Suc n))))) = c"
```
```   807     by (intro lim_imp_Liminf lim) simp
```
```   808   finally have l: "?l = c * ereal (inverse r)" by simp
```
```   809   from r have  l': "c * ereal (inverse r) > 1" by (cases c) (simp_all add: field_simps)
```
```   810   show "summable (\<lambda>n. f n * of_real r^n)"
```
```   811     by (rule summable_norm_cancel, rule ratio_test_convergence)
```
```   812        (insert r nz l l', auto elim!: eventually_mono)
```
```   813 next
```
```   814   fix r assume r: "0 < r" "ereal r > c"
```
```   815   let ?l = "limsup (\<lambda>n. ereal (norm (f n * of_real r ^ n) / norm (f (Suc n) * of_real r ^ Suc n)))"
```
```   816   have "?l = limsup (\<lambda>n. ereal (norm (f n) / (norm (f (Suc n)))) * ereal (inverse r))"
```
```   817     using r by (simp add: norm_mult norm_power divide_simps)
```
```   818   also from r have "\<dots> = limsup (\<lambda>n. ereal (norm (f n) / (norm (f (Suc n))))) * ereal (inverse r)"
```
```   819     by (intro Limsup_ereal_mult_right) simp_all
```
```   820   also have "limsup (\<lambda>n. ereal (norm (f n) / (norm (f (Suc n))))) = c"
```
```   821     by (intro lim_imp_Limsup lim) simp
```
```   822   finally have l: "?l = c * ereal (inverse r)" by simp
```
```   823   from r have  l': "c * ereal (inverse r) < 1" by (cases c) (simp_all add: field_simps)
```
```   824   show "\<not>summable (\<lambda>n. norm (f n * of_real r^n))"
```
```   825     by (rule ratio_test_divergence) (insert r nz l l', auto elim!: eventually_mono)
```
```   826 qed
```
```   827
```
```   828 lemma conv_radius_ratio_limit_ereal_nonzero:
```
```   829   fixes f :: "nat \<Rightarrow> 'a :: {banach,real_normed_div_algebra}"
```
```   830   assumes nz:  "c \<noteq> 0"
```
```   831   assumes lim: "(\<lambda>n. ereal (norm (f n) / norm (f (Suc n)))) \<longlonglongrightarrow> c"
```
```   832   shows   "conv_radius f = c"
```
```   833 proof (rule conv_radius_ratio_limit_ereal[OF _ lim], rule ccontr)
```
```   834   assume "\<not>eventually (\<lambda>n. f n \<noteq> 0) sequentially"
```
```   835   hence "frequently (\<lambda>n. f n = 0) sequentially" by (simp add: frequently_def)
```
```   836   hence "frequently (\<lambda>n. ereal (norm (f n) / norm (f (Suc n))) = 0) sequentially"
```
```   837     by (force elim!: frequently_elim1)
```
```   838   hence "c = 0" by (intro limit_frequently_eq[OF _ _ lim]) auto
```
```   839   with nz show False by contradiction
```
```   840 qed
```
```   841
```
```   842 lemma conv_radius_ratio_limit:
```
```   843   fixes f :: "nat \<Rightarrow> 'a :: {banach,real_normed_div_algebra}"
```
```   844   assumes "c' = ereal c"
```
```   845   assumes nz:  "eventually (\<lambda>n. f n \<noteq> 0) sequentially"
```
```   846   assumes lim: "(\<lambda>n. norm (f n) / norm (f (Suc n))) \<longlonglongrightarrow> c"
```
```   847   shows   "conv_radius f = c'"
```
```   848   using assms by (intro conv_radius_ratio_limit_ereal) simp_all
```
```   849
```
```   850 lemma conv_radius_ratio_limit_nonzero:
```
```   851   fixes f :: "nat \<Rightarrow> 'a :: {banach,real_normed_div_algebra}"
```
```   852   assumes "c' = ereal c"
```
```   853   assumes nz:  "c \<noteq> 0"
```
```   854   assumes lim: "(\<lambda>n. norm (f n) / norm (f (Suc n))) \<longlonglongrightarrow> c"
```
```   855   shows   "conv_radius f = c'"
```
```   856   using assms by (intro conv_radius_ratio_limit_ereal_nonzero) simp_all
```
```   857
```
```   858 lemma conv_radius_mult_power:
```
```   859   assumes "c \<noteq> (0 :: 'a :: {real_normed_div_algebra,banach})"
```
```   860   shows   "conv_radius (\<lambda>n. c ^ n * f n) = conv_radius f / ereal (norm c)"
```
```   861 proof -
```
```   862   have "limsup (\<lambda>n. ereal (root n (norm (c ^ n * f n)))) =
```
```   863           limsup (\<lambda>n. ereal (norm c) * ereal (root n (norm (f n))))"
```
```   864     using eventually_gt_at_top[of "0::nat"]
```
```   865     by (intro Limsup_eq)
```
```   866        (auto elim!: eventually_mono simp: norm_mult norm_power real_root_mult real_root_power)
```
```   867   also have "\<dots> = ereal (norm c) * limsup (\<lambda>n. ereal (root n (norm (f n))))"
```
```   868     using assms by (subst Limsup_ereal_mult_left[symmetric]) simp_all
```
```   869   finally have A: "limsup (\<lambda>n. ereal (root n (norm (c ^ n * f n)))) =
```
```   870                        ereal (norm c) * limsup (\<lambda>n. ereal (root n (norm (f n))))" .
```
```   871   show ?thesis using assms
```
```   872     apply (cases "limsup (\<lambda>n. ereal (root n (norm (f n)))) = 0")
```
```   873     apply (simp add: A conv_radius_def)
```
```   874     apply (unfold conv_radius_def A divide_ereal_def, simp add: mult.commute ereal_inverse_mult)
```
```   875     done
```
```   876 qed
```
```   877
```
```   878 lemma conv_radius_mult_power_right:
```
```   879   assumes "c \<noteq> (0 :: 'a :: {real_normed_div_algebra,banach})"
```
```   880   shows   "conv_radius (\<lambda>n. f n * c ^ n) = conv_radius f / ereal (norm c)"
```
```   881   using conv_radius_mult_power[OF assms, of f]
```
```   882   unfolding conv_radius_def by (simp add: mult.commute norm_mult)
```
```   883
```
```   884 lemma conv_radius_divide_power:
```
```   885   assumes "c \<noteq> (0 :: 'a :: {real_normed_div_algebra,banach})"
```
```   886   shows   "conv_radius (\<lambda>n. f n / c^n) = conv_radius f * ereal (norm c)"
```
```   887 proof -
```
```   888   from assms have "inverse c \<noteq> 0" by simp
```
```   889   from conv_radius_mult_power_right[OF this, of f] show ?thesis
```
```   890     by (simp add: divide_inverse divide_ereal_def assms norm_inverse power_inverse)
```
```   891 qed
```
```   892
```
```   893
```
```   894 lemma conv_radius_add_ge:
```
```   895   "min (conv_radius f) (conv_radius g) \<le>
```
```   896        conv_radius (\<lambda>x. f x + g x :: 'a :: {banach,real_normed_div_algebra})"
```
```   897   by (rule conv_radius_geI_ex')
```
```   898      (auto simp: algebra_simps intro!: summable_add summable_in_conv_radius)
```
```   899
```
```   900 lemma conv_radius_mult_ge:
```
```   901   fixes f g :: "nat \<Rightarrow> ('a :: {banach,real_normed_div_algebra})"
```
```   902   shows "conv_radius (\<lambda>x. \<Sum>i\<le>x. f i * g (x - i)) \<ge> min (conv_radius f) (conv_radius g)"
```
```   903 proof (rule conv_radius_geI_ex')
```
```   904   fix r assume r: "r > 0" "ereal r < min (conv_radius f) (conv_radius g)"
```
```   905   from r have "summable (\<lambda>n. (\<Sum>i\<le>n. (f i * of_real r^i) * (g (n - i) * of_real r^(n - i))))"
```
```   906     by (intro summable_Cauchy_product abs_summable_in_conv_radius) simp_all
```
```   907   thus "summable (\<lambda>n. (\<Sum>i\<le>n. f i * g (n - i)) * of_real r ^ n)"
```
```   908     by (simp add: algebra_simps of_real_def power_add [symmetric] scaleR_setsum_right)
```
```   909 qed
```
```   910
```
```   911 lemma le_conv_radius_iff:
```
```   912   fixes a :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}"
```
```   913   shows "r \<le> conv_radius a \<longleftrightarrow> (\<forall>x. norm (x-\<xi>) < r \<longrightarrow> summable (\<lambda>i. a i * (x - \<xi>) ^ i))"
```
```   914 apply (intro iffI allI impI summable_in_conv_radius conv_radius_geI_ex)
```
```   915 apply (meson less_ereal.simps(1) not_le order_trans)
```
```   916 apply (rule_tac x="of_real ra" in exI, simp)
```
```   917 apply (metis abs_of_nonneg add_diff_cancel_left' less_eq_real_def norm_of_real)
```
```   918 done
```
```   919
```
```   920 end
```