(* Title : Series.thy
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
Converted to Isar and polished by lcp
Converted to sum and polished yet more by TNN
Additional contributions by Jeremy Avigad
*)
section \<open>Infinite Series\<close>
theory Series
imports Limits Inequalities
begin
subsection \<open>Definition of infinite summability\<close>
definition sums :: "(nat \ 'a::{topological_space, comm_monoid_add}) \ 'a \ bool"
(infixr "sums" 80)
where "f sums s \ (\n. \i s"
definition summable :: "(nat \ 'a::{topological_space, comm_monoid_add}) \ bool"
where "summable f \ (\s. f sums s)"
definition suminf :: "(nat \ 'a::{topological_space, comm_monoid_add}) \ 'a"
(binder "\" 10)
where "suminf f = (THE s. f sums s)"
text\<open>Variants of the definition\<close>
lemma sums_def': "f sums s \ (\n. \i = 0..n. f i) \ s"
unfolding sums_def
apply (subst filterlim_sequentially_Suc [symmetric])
apply (simp only: lessThan_Suc_atMost atLeast0AtMost)
done
lemma sums_def_le: "f sums s \ (\n. \i\n. f i) \ s"
by (simp add: sums_def' atMost_atLeast0)
lemma bounded_imp_summable:
fixes a :: "nat \ 'a::{conditionally_complete_linorder,linorder_topology,linordered_comm_semiring_strict}"
assumes 0: "\n. a n \ 0" and bounded: "\n. (\k\n. a k) \ B"
shows "summable a"
proof -
have "bdd_above (range(\n. \k\n. a k))"
by (meson bdd_aboveI2 bounded)
moreover have "incseq (\n. \k\n. a k)"
by (simp add: mono_def "0" sum_mono2)
ultimately obtain s where "(\n. \k\n. a k) \ s"
using LIMSEQ_incseq_SUP by blast
then show ?thesis
by (auto simp: sums_def_le summable_def)
qed
subsection \<open>Infinite summability on topological monoids\<close>
lemma sums_subst[trans]: "f = g \ g sums z \ f sums z"
by simp
lemma sums_cong: "(\n. f n = g n) \ f sums c \ g sums c"
by (drule ext) simp
lemma sums_summable: "f sums l \ summable f"
by (simp add: sums_def summable_def, blast)
lemma summable_iff_convergent: "summable f \ convergent (\n. \i
by (simp add: summable_def sums_def convergent_def)
lemma summable_iff_convergent': "summable f \ convergent (\n. sum f {..n})"
by (simp_all only: summable_iff_convergent convergent_def
lessThan_Suc_atMost [symmetric] filterlim_sequentially_Suc[of "\n. sum f {..
lemma suminf_eq_lim: "suminf f = lim (\n. \i
by (simp add: suminf_def sums_def lim_def)
lemma sums_zero[simp, intro]: "(\n. 0) sums 0"
unfolding sums_def by simp
lemma summable_zero[simp, intro]: "summable (\n. 0)"
by (rule sums_zero [THEN sums_summable])
lemma sums_group: "f sums s \ 0 < k \ (\n. sum f {n * k ..< n * k + k}) sums s"
apply (simp only: sums_def sum.nat_group tendsto_def eventually_sequentially)
apply (erule all_forward imp_forward exE| assumption)+
apply (rule_tac x="N" in exI)
by (metis le_square mult.commute mult.left_neutral mult_le_cancel2 mult_le_mono)
lemma suminf_cong: "(\n. f n = g n) \ suminf f = suminf g"
by (rule arg_cong[of f g], rule ext) simp
lemma summable_cong:
fixes f g :: "nat \ 'a::real_normed_vector"
assumes "eventually (\x. f x = g x) sequentially"
shows "summable f = summable g"
proof -
from assms obtain N where N: "\n\N. f n = g n"
by (auto simp: eventually_at_top_linorder)
define C where "C = (\k
from eventually_ge_at_top[of N]
have "eventually (\n. sum f {..
proof eventually_elim
case (elim n)
then have "{.. {N..
by auto
also have "sum f ... = sum f {..
by (intro sum.union_disjoint) auto
also from N have "sum f {N..
by (intro sum.cong) simp_all
also have "sum f {..
unfolding C_def by (simp add: algebra_simps sum_subtractf)
also have "sum g {.. {N..
by (intro sum.union_disjoint [symmetric]) auto
also from elim have "{.. {N..
by auto
finally show "sum f {.. .
qed
from convergent_cong[OF this] show ?thesis
by (simp add: summable_iff_convergent convergent_add_const_iff)
qed
lemma sums_finite:
assumes [simp]: "finite N"
and f: "\n. n \ N \ f n = 0"
shows "f sums (\n\N. f n)"
proof -
have eq: "sum f {.. for n
by (rule sum.mono_neutral_right) (auto simp: add_increasing less_Suc_eq_le f)
show ?thesis
unfolding sums_def
by (rule LIMSEQ_offset[of _ "Suc (Max N)"])
(simp add: eq atLeast0LessThan del: add_Suc_right)
qed
corollary sums_0: "(\n. f n = 0) \ (f sums 0)"
by (metis (no_types) finite.emptyI sum.empty sums_finite)
lemma summable_finite: "finite N \ (\n. n \ N \ f n = 0) \ summable f"
by (rule sums_summable) (rule sums_finite)
lemma sums_If_finite_set: "finite A \ (\r. if r \ A then f r else 0) sums (\r\A. f r)"
using sums_finite[of A "(\r. if r \ A then f r else 0)"] by simp
lemma summable_If_finite_set[simp, intro]: "finite A \ summable (\r. if r \ A then f r else 0)"
by (rule sums_summable) (rule sums_If_finite_set)
lemma sums_If_finite: "finite {r. P r} \ (\r. if P r then f r else 0) sums (\r | P r. f r)"
using sums_If_finite_set[of "{r. P r}"] by simp
lemma summable_If_finite[simp, intro]: "finite {r. P r} \ summable (\r. if P r then f r else 0)"
by (rule sums_summable) (rule sums_If_finite)
lemma sums_single: "(\r. if r = i then f r else 0) sums f i"
using sums_If_finite[of "\r. r = i"] by simp
lemma summable_single[simp, intro]: "summable (\r. if r = i then f r else 0)"
by (rule sums_summable) (rule sums_single)
context
fixes f :: "nat \ 'a::{t2_space,comm_monoid_add}"
begin
lemma summable_sums[intro]: "summable f \ f sums (suminf f)"
by (simp add: summable_def sums_def suminf_def)
(metis convergent_LIMSEQ_iff convergent_def lim_def)
lemma summable_LIMSEQ: "summable f \ (\n. \i suminf f"
by (rule summable_sums [unfolded sums_def])
lemma summable_LIMSEQ': "summable f \ (\n. \i\n. f i) \ suminf f"
using sums_def_le by blast
lemma sums_unique: "f sums s \ s = suminf f"
by (metis limI suminf_eq_lim sums_def)
lemma sums_iff: "f sums x \ summable f \ suminf f = x"
by (metis summable_sums sums_summable sums_unique)
lemma summable_sums_iff: "summable f \ f sums suminf f"
by (auto simp: sums_iff summable_sums)
lemma sums_unique2: "f sums a \ f sums b \ a = b"
for a b :: 'a
by (simp add: sums_iff)
lemma sums_Uniq: "\\<^sub>\\<^sub>1a. f sums a"
for a b :: 'a
by (simp add: sums_unique2 Uniq_def)
lemma suminf_finite:
assumes N: "finite N"
and f: "\n. n \ N \ f n = 0"
shows "suminf f = (\n\N. f n)"
using sums_finite[OF assms, THEN sums_unique] by simp
end
lemma suminf_zero[simp]: "suminf (\n. 0::'a::{t2_space, comm_monoid_add}) = 0"
by (rule sums_zero [THEN sums_unique, symmetric])
subsection \<open>Infinite summability on ordered, topological monoids\<close>
lemma sums_le: "(\n. f n \ g n) \ f sums s \ g sums t \ s \ t"
for f g :: "nat \ 'a::{ordered_comm_monoid_add,linorder_topology}"
by (rule LIMSEQ_le) (auto intro: sum_mono simp: sums_def)
context
fixes f :: "nat \ 'a::{ordered_comm_monoid_add,linorder_topology}"
begin
lemma suminf_le: "(\n. f n \ g n) \ summable f \ summable g \ suminf f \ suminf g"
using sums_le by blast
lemma sum_le_suminf:
shows "summable f \ finite I \ (\n. n \- I \ 0 \ f n) \ sum f I \ suminf f"
by (rule sums_le[OF _ sums_If_finite_set summable_sums]) auto
lemma suminf_nonneg: "summable f \ (\n. 0 \ f n) \ 0 \ suminf f"
using sum_le_suminf by force
lemma suminf_le_const: "summable f \ (\n. sum f {.. x) \ suminf f \ x"
by (metis LIMSEQ_le_const2 summable_LIMSEQ)
lemma suminf_eq_zero_iff:
assumes "summable f" and pos: "\n. 0 \ f n"
shows "suminf f = 0 \ (\n. f n = 0)"
proof
assume "suminf f = 0"
then have f: "(\n. \i 0"
using summable_LIMSEQ[of f] assms by simp
then have "\i. (\n\{i}. f n) \ 0"
proof (rule LIMSEQ_le_const)
show "\N. \n\N. (\n\{i}. f n) \ sum f {..
using pos by (intro exI[of _ "Suc i"] allI impI sum_mono2) auto
qed
with pos show "\n. f n = 0"
by (auto intro!: antisym)
qed (metis suminf_zero fun_eq_iff)
lemma suminf_pos_iff: "summable f \ (\n. 0 \ f n) \ 0 < suminf f \ (\i. 0 < f i)"
using sum_le_suminf[of "{}"] suminf_eq_zero_iff by (simp add: less_le)
lemma suminf_pos2:
assumes "summable f" "\n. 0 \ f n" "0 < f i"
shows "0 < suminf f"
proof -
have "0 < (\n
using assms by (intro sum_pos2[where i=i]) auto
also have "\ \ suminf f"
using assms by (intro sum_le_suminf) auto
finally show ?thesis .
qed
lemma suminf_pos: "summable f \ (\n. 0 < f n) \ 0 < suminf f"
by (intro suminf_pos2[where i=0]) (auto intro: less_imp_le)
end
context
fixes f :: "nat \ 'a::{ordered_cancel_comm_monoid_add,linorder_topology}"
begin
lemma sum_less_suminf2:
"summable f \ (\m. m\n \ 0 \ f m) \ n \ i \ 0 < f i \ sum f {..
using sum_le_suminf[of f "{..< Suc i}"]
and add_strict_increasing[of "f i" "sum f {.. "sum f {..]
and sum_mono2[of "{.. "{.. f]
by (auto simp: less_imp_le ac_simps)
lemma sum_less_suminf: "summable f \ (\m. m\n \ 0 < f m) \ sum f {..
using sum_less_suminf2[of n n] by (simp add: less_imp_le)
end
lemma summableI_nonneg_bounded:
fixes f :: "nat \ 'a::{ordered_comm_monoid_add,linorder_topology,conditionally_complete_linorder}"
assumes pos[simp]: "\n. 0 \ f n"
and le: "\n. (\i x"
shows "summable f"
unfolding summable_def sums_def [abs_def]
proof (rule exI LIMSEQ_incseq_SUP)+
show "bdd_above (range (\n. sum f {..
using le by (auto simp: bdd_above_def)
show "incseq (\n. sum f {..
by (auto simp: mono_def intro!: sum_mono2)
qed
lemma summableI[intro, simp]: "summable f"
for f :: "nat \ 'a::{canonically_ordered_monoid_add,linorder_topology,complete_linorder}"
by (intro summableI_nonneg_bounded[where x=top] zero_le top_greatest)
lemma suminf_eq_SUP_real:
assumes X: "summable X" "\i. 0 \ X i" shows "suminf X = (SUP i. \n
by (intro LIMSEQ_unique[OF summable_LIMSEQ] X LIMSEQ_incseq_SUP)
(auto intro!: bdd_aboveI2[where M="\i. X i"] sum_le_suminf X monoI sum_mono2)
subsection \<open>Infinite summability on topological monoids\<close>
context
fixes f g :: "nat \ 'a::{t2_space,topological_comm_monoid_add}"
begin
lemma sums_Suc:
assumes "(\n. f (Suc n)) sums l"
shows "f sums (l + f 0)"
proof -
have "(\n. (\i l + f 0"
using assms by (auto intro!: tendsto_add simp: sums_def)
moreover have "(\ii
unfolding lessThan_Suc_eq_insert_0
by (simp add: ac_simps sum.atLeast1_atMost_eq image_Suc_lessThan)
ultimately show ?thesis
by (auto simp: sums_def simp del: sum.lessThan_Suc intro: filterlim_sequentially_Suc[THEN iffD1])
qed
lemma sums_add: "f sums a \ g sums b \ (\n. f n + g n) sums (a + b)"
unfolding sums_def by (simp add: sum.distrib tendsto_add)
lemma summable_add: "summable f \ summable g \ summable (\n. f n + g n)"
unfolding summable_def by (auto intro: sums_add)
lemma suminf_add: "summable f \ summable g \ suminf f + suminf g = (\n. f n + g n)"
by (intro sums_unique sums_add summable_sums)
end
context
fixes f :: "'i \ nat \ 'a::{t2_space,topological_comm_monoid_add}"
and I :: "'i set"
begin
lemma sums_sum: "(\i. i \ I \ (f i) sums (x i)) \ (\n. \i\I. f i n) sums (\i\I. x i)"
by (induct I rule: infinite_finite_induct) (auto intro!: sums_add)
lemma suminf_sum: "(\i. i \ I \ summable (f i)) \ (\n. \i\I. f i n) = (\i\I. \n. f i n)"
using sums_unique[OF sums_sum, OF summable_sums] by simp
lemma summable_sum: "(\i. i \ I \ summable (f i)) \ summable (\n. \i\I. f i n)"
using sums_summable[OF sums_sum[OF summable_sums]] .
end
lemma sums_If_finite_set':
fixes f g :: "nat \ 'a::{t2_space,topological_ab_group_add}"
assumes "g sums S" and "finite A" and "S' = S + (\n\A. f n - g n)"
shows "(\n. if n \ A then f n else g n) sums S'"
proof -
have "(\n. g n + (if n \ A then f n - g n else 0)) sums (S + (\n\A. f n - g n))"
by (intro sums_add assms sums_If_finite_set)
also have "(\n. g n + (if n \ A then f n - g n else 0)) = (\n. if n \ A then f n else g n)"
by (simp add: fun_eq_iff)
finally show ?thesis using assms by simp
qed
subsection \<open>Infinite summability on real normed vector spaces\<close>
context
fixes f :: "nat \ 'a::real_normed_vector"
begin
lemma sums_Suc_iff: "(\n. f (Suc n)) sums s \ f sums (s + f 0)"
proof -
have "f sums (s + f 0) \ (\i. \j s + f 0"
by (subst filterlim_sequentially_Suc) (simp add: sums_def)
also have "\ \ (\i. (\j s + f 0"
by (simp add: ac_simps lessThan_Suc_eq_insert_0 image_Suc_lessThan sum.atLeast1_atMost_eq)
also have "\ \ (\n. f (Suc n)) sums s"
proof
assume "(\i. (\j s + f 0"
with tendsto_add[OF this tendsto_const, of "- f 0"] show "(\i. f (Suc i)) sums s"
by (simp add: sums_def)
qed (auto intro: tendsto_add simp: sums_def)
finally show ?thesis ..
qed
lemma summable_Suc_iff: "summable (\n. f (Suc n)) = summable f"
proof
assume "summable f"
then have "f sums suminf f"
by (rule summable_sums)
then have "(\n. f (Suc n)) sums (suminf f - f 0)"
by (simp add: sums_Suc_iff)
then show "summable (\n. f (Suc n))"
unfolding summable_def by blast
qed (auto simp: sums_Suc_iff summable_def)
lemma sums_Suc_imp: "f 0 = 0 \ (\n. f (Suc n)) sums s \ (\n. f n) sums s"
using sums_Suc_iff by simp
end
context (* Separate contexts are necessary to allow general use of the results above, here. *)
fixes f :: "nat \ 'a::real_normed_vector"
begin
lemma sums_diff: "f sums a \ g sums b \ (\n. f n - g n) sums (a - b)"
unfolding sums_def by (simp add: sum_subtractf tendsto_diff)
lemma summable_diff: "summable f \ summable g \ summable (\n. f n - g n)"
unfolding summable_def by (auto intro: sums_diff)
lemma suminf_diff: "summable f \ summable g \ suminf f - suminf g = (\n. f n - g n)"
by (intro sums_unique sums_diff summable_sums)
lemma sums_minus: "f sums a \ (\n. - f n) sums (- a)"
unfolding sums_def by (simp add: sum_negf tendsto_minus)
lemma summable_minus: "summable f \ summable (\n. - f n)"
unfolding summable_def by (auto intro: sums_minus)
lemma suminf_minus: "summable f \ (\n. - f n) = - (\n. f n)"
by (intro sums_unique [symmetric] sums_minus summable_sums)
lemma sums_iff_shift: "(\i. f (i + n)) sums s \ f sums (s + (\i
proof (induct n arbitrary: s)
case 0
then show ?case by simp
next
case (Suc n)
then have "(\i. f (Suc i + n)) sums s \ (\i. f (i + n)) sums (s + f n)"
by (subst sums_Suc_iff) simp
with Suc show ?case
by (simp add: ac_simps)
qed
corollary sums_iff_shift': "(\i. f (i + n)) sums (s - (\i f sums s"
by (simp add: sums_iff_shift)
lemma sums_zero_iff_shift:
assumes "\i. i < n \ f i = 0"
shows "(\i. f (i+n)) sums s \ (\i. f i) sums s"
by (simp add: assms sums_iff_shift)
lemma summable_iff_shift [simp]: "summable (\n. f (n + k)) \ summable f"
by (metis diff_add_cancel summable_def sums_iff_shift [abs_def])
lemma sums_split_initial_segment: "f sums s \ (\i. f (i + n)) sums (s - (\i
by (simp add: sums_iff_shift)
lemma summable_ignore_initial_segment: "summable f \ summable (\n. f(n + k))"
by (simp add: summable_iff_shift)
lemma suminf_minus_initial_segment: "summable f \ (\n. f (n + k)) = (\n. f n) - (\i
by (rule sums_unique[symmetric]) (auto simp: sums_iff_shift)
lemma suminf_split_initial_segment: "summable f \ suminf f = (\n. f(n + k)) + (\i
by (auto simp add: suminf_minus_initial_segment)
lemma suminf_split_head: "summable f \ (\n. f (Suc n)) = suminf f - f 0"
using suminf_split_initial_segment[of 1] by simp
lemma suminf_exist_split:
fixes r :: real
assumes "0 < r" and "summable f"
shows "\N. \n\N. norm (\i. f (i + n)) < r"
proof -
from LIMSEQ_D[OF summable_LIMSEQ[OF \<open>summable f\<close>] \<open>0 < r\<close>]
obtain N :: nat where "\ n \ N. norm (sum f {..
by auto
then show ?thesis
by (auto simp: norm_minus_commute suminf_minus_initial_segment[OF \<open>summable f\<close>])
qed
lemma summable_LIMSEQ_zero:
assumes "summable f" shows "f \ 0"
proof -
have "Cauchy (\n. sum f {..
using LIMSEQ_imp_Cauchy assms summable_LIMSEQ by blast
then show ?thesis
unfolding Cauchy_iff LIMSEQ_iff
by (metis add.commute add_diff_cancel_right' diff_zero le_SucI sum.lessThan_Suc)
qed
lemma summable_imp_convergent: "summable f \ convergent f"
by (force dest!: summable_LIMSEQ_zero simp: convergent_def)
lemma summable_imp_Bseq: "summable f \ Bseq f"
by (simp add: convergent_imp_Bseq summable_imp_convergent)
end
lemma summable_minus_iff: "summable (\n. - f n) \ summable f"
for f :: "nat \ 'a::real_normed_vector"
by (auto dest: summable_minus) (* used two ways, hence must be outside the context above *)
lemma (in bounded_linear) sums: "(\n. X n) sums a \ (\n. f (X n)) sums (f a)"
unfolding sums_def by (drule tendsto) (simp only: sum)
lemma (in bounded_linear) summable: "summable (\n. X n) \ summable (\n. f (X n))"
unfolding summable_def by (auto intro: sums)
lemma (in bounded_linear) suminf: "summable (\n. X n) \ f (\n. X n) = (\n. f (X n))"
by (intro sums_unique sums summable_sums)
lemmas sums_of_real = bounded_linear.sums [OF bounded_linear_of_real]
lemmas summable_of_real = bounded_linear.summable [OF bounded_linear_of_real]
lemmas suminf_of_real = bounded_linear.suminf [OF bounded_linear_of_real]
lemmas sums_scaleR_left = bounded_linear.sums[OF bounded_linear_scaleR_left]
lemmas summable_scaleR_left = bounded_linear.summable[OF bounded_linear_scaleR_left]
lemmas suminf_scaleR_left = bounded_linear.suminf[OF bounded_linear_scaleR_left]
lemmas sums_scaleR_right = bounded_linear.sums[OF bounded_linear_scaleR_right]
lemmas summable_scaleR_right = bounded_linear.summable[OF bounded_linear_scaleR_right]
lemmas suminf_scaleR_right = bounded_linear.suminf[OF bounded_linear_scaleR_right]
lemma summable_const_iff: "summable (\_. c) \ c = 0"
for c :: "'a::real_normed_vector"
proof -
have "\ summable (\_. c)" if "c \ 0"
proof -
from that have "filterlim (\n. of_nat n * norm c) at_top sequentially"
by (subst mult.commute)
(auto intro!: filterlim_tendsto_pos_mult_at_top filterlim_real_sequentially)
then have "\ convergent (\n. norm (\k
by (intro filterlim_at_infinity_imp_not_convergent filterlim_at_top_imp_at_infinity)
(simp_all add: sum_constant_scaleR)
then show ?thesis
unfolding summable_iff_convergent using convergent_norm by blast
qed
then show ?thesis by auto
qed
subsection \<open>Infinite summability on real normed algebras\<close>
context
fixes f :: "nat \ 'a::real_normed_algebra"
begin
lemma sums_mult: "f sums a \ (\n. c * f n) sums (c * a)"
by (rule bounded_linear.sums [OF bounded_linear_mult_right])
lemma summable_mult: "summable f \ summable (\n. c * f n)"
by (rule bounded_linear.summable [OF bounded_linear_mult_right])
lemma suminf_mult: "summable f \ suminf (\n. c * f n) = c * suminf f"
by (rule bounded_linear.suminf [OF bounded_linear_mult_right, symmetric])
lemma sums_mult2: "f sums a \ (\n. f n * c) sums (a * c)"
by (rule bounded_linear.sums [OF bounded_linear_mult_left])
lemma summable_mult2: "summable f \ summable (\n. f n * c)"
by (rule bounded_linear.summable [OF bounded_linear_mult_left])
lemma suminf_mult2: "summable f \ suminf f * c = (\n. f n * c)"
by (rule bounded_linear.suminf [OF bounded_linear_mult_left])
end
lemma sums_mult_iff:
fixes f :: "nat \ 'a::{real_normed_algebra,field}"
assumes "c \ 0"
shows "(\n. c * f n) sums (c * d) \ f sums d"
using sums_mult[of f d c] sums_mult[of "\n. c * f n" "c * d" "inverse c"]
by (force simp: field_simps assms)
lemma sums_mult2_iff:
fixes f :: "nat \ 'a::{real_normed_algebra,field}"
assumes "c \ 0"
shows "(\n. f n * c) sums (d * c) \ f sums d"
using sums_mult_iff[OF assms, of f d] by (simp add: mult.commute)
lemma sums_of_real_iff:
"(\n. of_real (f n) :: 'a::real_normed_div_algebra) sums of_real c \ f sums c"
by (simp add: sums_def of_real_sum[symmetric] tendsto_of_real_iff del: of_real_sum)
subsection \<open>Infinite summability on real normed fields\<close>
context
fixes c :: "'a::real_normed_field"
begin
lemma sums_divide: "f sums a \ (\n. f n / c) sums (a / c)"
by (rule bounded_linear.sums [OF bounded_linear_divide])
lemma summable_divide: "summable f \ summable (\n. f n / c)"
by (rule bounded_linear.summable [OF bounded_linear_divide])
lemma suminf_divide: "summable f \ suminf (\n. f n / c) = suminf f / c"
by (rule bounded_linear.suminf [OF bounded_linear_divide, symmetric])
lemma summable_inverse_divide: "summable (inverse \ f) \ summable (\n. c / f n)"
by (auto dest: summable_mult [of _ c] simp: field_simps)
lemma sums_mult_D: "(\n. c * f n) sums a \ c \ 0 \ f sums (a/c)"
using sums_mult_iff by fastforce
lemma summable_mult_D: "summable (\n. c * f n) \ c \ 0 \ summable f"
by (auto dest: summable_divide)
text \<open>Sum of a geometric progression.\<close>
lemma geometric_sums:
assumes "norm c < 1"
shows "(\n. c^n) sums (1 / (1 - c))"
proof -
have neq_0: "c - 1 \ 0"
using assms by auto
then have "(\n. c ^ n / (c - 1) - 1 / (c - 1)) \ 0 / (c - 1) - 1 / (c - 1)"
by (intro tendsto_intros assms)
then have "(\n. (c ^ n - 1) / (c - 1)) \ 1 / (1 - c)"
by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib)
with neq_0 show "(\n. c ^ n) sums (1 / (1 - c))"
by (simp add: sums_def geometric_sum)
qed
lemma summable_geometric: "norm c < 1 \ summable (\n. c^n)"
by (rule geometric_sums [THEN sums_summable])
lemma suminf_geometric: "norm c < 1 \ suminf (\n. c^n) = 1 / (1 - c)"
by (rule sums_unique[symmetric]) (rule geometric_sums)
lemma summable_geometric_iff [simp]: "summable (\n. c ^ n) \ norm c < 1"
proof
assume "summable (\n. c ^ n :: 'a :: real_normed_field)"
then have "(\n. norm c ^ n) \ 0"
by (simp add: norm_power [symmetric] tendsto_norm_zero_iff summable_LIMSEQ_zero)
from order_tendstoD(2)[OF this zero_less_one] obtain n where "norm c ^ n < 1"
by (auto simp: eventually_at_top_linorder)
then show "norm c < 1" using one_le_power[of "norm c" n]
by (cases "norm c \ 1") (linarith, simp)
qed (rule summable_geometric)
end
text \<open>Biconditional versions for constant factors\<close>
context
fixes c :: "'a::real_normed_field"
begin
lemma summable_cmult_iff [simp]: "summable (\n. c * f n) \ c=0 \ summable f"
proof -
have "\summable (\n. c * f n); c \ 0\ \ summable f"
using summable_mult_D by blast
then show ?thesis
by (auto simp: summable_mult)
qed
lemma summable_divide_iff [simp]: "summable (\n. f n / c) \ c=0 \ summable f"
proof -
have "\summable (\n. f n / c); c \ 0\ \ summable f"
by (auto dest: summable_divide [where c = "1/c"])
then show ?thesis
by (auto simp: summable_divide)
qed
end
lemma power_half_series: "(\n. (1/2::real)^Suc n) sums 1"
proof -
have 2: "(\n. (1/2::real)^n) sums 2"
using geometric_sums [of "1/2::real"] by auto
have "(\n. (1/2::real)^Suc n) = (\n. (1 / 2) ^ n / 2)"
by (simp add: mult.commute)
then show ?thesis
using sums_divide [OF 2, of 2] by simp
qed
subsection \<open>Telescoping\<close>
lemma telescope_sums:
fixes c :: "'a::real_normed_vector"
assumes "f \ c"
shows "(\n. f (Suc n) - f n) sums (c - f 0)"
unfolding sums_def
proof (subst filterlim_sequentially_Suc [symmetric])
have "(\n. \kn. f (Suc n) - f 0)"
by (simp add: lessThan_Suc_atMost atLeast0AtMost [symmetric] sum_Suc_diff)
also have "\ \ c - f 0"
by (intro tendsto_diff LIMSEQ_Suc[OF assms] tendsto_const)
finally show "(\n. \n c - f 0" .
qed
lemma telescope_sums':
fixes c :: "'a::real_normed_vector"
assumes "f \ c"
shows "(\n. f n - f (Suc n)) sums (f 0 - c)"
using sums_minus[OF telescope_sums[OF assms]] by (simp add: algebra_simps)
lemma telescope_summable:
fixes c :: "'a::real_normed_vector"
assumes "f \ c"
shows "summable (\n. f (Suc n) - f n)"
using telescope_sums[OF assms] by (simp add: sums_iff)
lemma telescope_summable':
fixes c :: "'a::real_normed_vector"
assumes "f \ c"
shows "summable (\n. f n - f (Suc n))"
using summable_minus[OF telescope_summable[OF assms]] by (simp add: algebra_simps)
subsection \<open>Infinite summability on Banach spaces\<close>
text \<open>Cauchy-type criterion for convergence of series (c.f. Harrison).\<close>
lemma summable_Cauchy: "summable f \ (\e>0. \N. \m\N. \n. norm (sum f {m..
for f :: "nat \ 'a::banach"
proof
assume f: "summable f"
show ?rhs
proof clarify
fix e :: real
assume "0 < e"
then obtain M where M: "\m n. \m\M; n\M\ \ norm (sum f {..
using f by (force simp add: summable_iff_convergent Cauchy_convergent_iff [symmetric] Cauchy_iff)
have "norm (sum f {m.. if "m \ M" for m n
proof (cases m n rule: linorder_class.le_cases)
assume "m \ n"
then show ?thesis
by (metis (mono_tags, hide_lams) M atLeast0LessThan order_trans sum_diff_nat_ivl that zero_le)
next
assume "n \ m"
then show ?thesis
by (simp add: \<open>0 < e\<close>)
qed
then show "\N. \m\N. \n. norm (sum f {m..
by blast
qed
next
assume r: ?rhs
then show "summable f"
unfolding summable_iff_convergent Cauchy_convergent_iff [symmetric] Cauchy_iff
proof clarify
fix e :: real
assume "0 < e"
with r obtain N where N: "\m n. m \ N \ norm (sum f {m..
by blast
have "norm (sum f {.. if "m\N" "n\N" for m n
proof (cases m n rule: linorder_class.le_cases)
assume "m \ n"
then show ?thesis
by (metis Groups_Big.sum_diff N finite_lessThan lessThan_minus_lessThan lessThan_subset_iff norm_minus_commute \<open>m\<ge>N\<close>)
next
assume "n \ m"
then show ?thesis
by (metis Groups_Big.sum_diff N finite_lessThan lessThan_minus_lessThan lessThan_subset_iff \<open>n\<ge>N\<close>)
qed
then show "\M. \m\M. \n\M. norm (sum f {..
by blast
qed
qed
lemma summable_Cauchy':
fixes f :: "nat \ 'a :: banach"
assumes "eventually (\m. \n\m. norm (sum f {m.. g m) sequentially"
assumes "filterlim g (nhds 0) sequentially"
shows "summable f"
proof (subst summable_Cauchy, intro allI impI, goal_cases)
case (1 e)
from order_tendstoD(2)[OF assms(2) this] and assms(1)
have "eventually (\m. \n. norm (sum f {m..
proof eventually_elim
case (elim m)
show ?case
proof
fix n
from elim show "norm (sum f {m..
by (cases "n \ m") auto
qed
qed
thus ?case by (auto simp: eventually_at_top_linorder)
qed
context
fixes f :: "nat \ 'a::banach"
begin
text \<open>Absolute convergence imples normal convergence.\<close>
lemma summable_norm_cancel: "summable (\n. norm (f n)) \ summable f"
unfolding summable_Cauchy
apply (erule all_forward imp_forward ex_forward | assumption)+
apply (fastforce simp add: order_le_less_trans [OF norm_sum] order_le_less_trans [OF abs_ge_self])
done
lemma summable_norm: "summable (\n. norm (f n)) \ norm (suminf f) \ (\n. norm (f n))"
by (auto intro: LIMSEQ_le tendsto_norm summable_norm_cancel summable_LIMSEQ norm_sum)
text \<open>Comparison tests.\<close>
lemma summable_comparison_test:
assumes fg: "\N. \n\N. norm (f n) \ g n" and g: "summable g"
shows "summable f"
proof -
obtain N where N: "\n. n\N \ norm (f n) \ g n"
using assms by blast
show ?thesis
proof (clarsimp simp add: summable_Cauchy)
fix e :: real
assume "0 < e"
then obtain Ng where Ng: "\m n. m \ Ng \ norm (sum g {m..
using g by (fastforce simp: summable_Cauchy)
with N have "norm (sum f {m.. if "m\max N Ng" for m n
proof -
have "norm (sum f {m.. sum g {m..
using N that by (force intro: sum_norm_le)
also have "... \ norm (sum g {m..
by simp
also have "... < e"
using Ng that by auto
finally show ?thesis .
qed
then show "\N. \m\N. \n. norm (sum f {m..
by blast
qed
qed
lemma summable_comparison_test_ev:
"eventually (\n. norm (f n) \ g n) sequentially \ summable g \ summable f"
by (rule summable_comparison_test) (auto simp: eventually_at_top_linorder)
text \<open>A better argument order.\<close>
lemma summable_comparison_test': "summable g \ (\n. n \ N \ norm (f n) \ g n) \ summable f"
by (rule summable_comparison_test) auto
subsection \<open>The Ratio Test\<close>
lemma summable_ratio_test:
assumes "c < 1" "\n. n \ N \ norm (f (Suc n)) \ c * norm (f n)"
shows "summable f"
proof (cases "0 < c")
case True
show "summable f"
proof (rule summable_comparison_test)
show "\N'. \n\N'. norm (f n) \ (norm (f N) / (c ^ N)) * c ^ n"
proof (intro exI allI impI)
fix n
assume "N \ n"
then show "norm (f n) \ (norm (f N) / (c ^ N)) * c ^ n"
proof (induct rule: inc_induct)
case base
with True show ?case by simp
next
case (step m)
have "norm (f (Suc m)) / c ^ Suc m * c ^ n \ norm (f m) / c ^ m * c ^ n"
using \<open>0 < c\<close> \<open>c < 1\<close> assms(2)[OF \<open>N \<le> m\<close>] by (simp add: field_simps)
with step show ?case by simp
qed
qed
show "summable (\n. norm (f N) / c ^ N * c ^ n)"
using \<open>0 < c\<close> \<open>c < 1\<close> by (intro summable_mult summable_geometric) simp
qed
next
case False
have "f (Suc n) = 0" if "n \ N" for n
proof -
from that have "norm (f (Suc n)) \ c * norm (f n)"
by (rule assms(2))
also have "\ \ 0"
using False by (simp add: not_less mult_nonpos_nonneg)
finally show ?thesis
by auto
qed
then show "summable f"
by (intro sums_summable[OF sums_finite, of "{.. Suc N}"]) (auto simp: not_le Suc_less_eq2)
qed
end
text \<open>Relations among convergence and absolute convergence for power series.\<close>
lemma Abel_lemma:
fixes a :: "nat \ 'a::real_normed_vector"
assumes r: "0 \ r"
and r0: "r < r0"
and M: "\n. norm (a n) * r0^n \ M"
shows "summable (\n. norm (a n) * r^n)"
proof (rule summable_comparison_test')
show "summable (\n. M * (r / r0) ^ n)"
using assms by (auto simp add: summable_mult summable_geometric)
show "norm (norm (a n) * r ^ n) \ M * (r / r0) ^ n" for n
using r r0 M [of n] dual_order.order_iff_strict
by (fastforce simp add: abs_mult field_simps)
qed
text \<open>Summability of geometric series for real algebras.\<close>
lemma complete_algebra_summable_geometric:
fixes x :: "'a::{real_normed_algebra_1,banach}"
assumes "norm x < 1"
shows "summable (\n. x ^ n)"
proof (rule summable_comparison_test)
show "\N. \n\N. norm (x ^ n) \ norm x ^ n"
by (simp add: norm_power_ineq)
from assms show "summable (\n. norm x ^ n)"
by (simp add: summable_geometric)
qed
subsection \<open>Cauchy Product Formula\<close>
text \<open>
Proof based on Analysis WebNotes: Chapter 07, Class 41
\<^url>\<open>http://www.math.unl.edu/~webnotes/classes/class41/prp77.htm\<close>
\<close>
lemma Cauchy_product_sums:
fixes a b :: "nat \ 'a::{real_normed_algebra,banach}"
assumes a: "summable (\k. norm (a k))"
and b: "summable (\k. norm (b k))"
shows "(\k. \i\k. a i * b (k - i)) sums ((\k. a k) * (\k. b k))"
proof -
let ?S1 = "\n::nat. {.. {..
let ?S2 = "\n::nat. {(i,j). i + j < n}"
have S1_mono: "\m n. m \ n \ ?S1 m \ ?S1 n" by auto
have S2_le_S1: "\n. ?S2 n \ ?S1 n" by auto
have S1_le_S2: "\n. ?S1 (n div 2) \ ?S2 n" by auto
have finite_S1: "\n. finite (?S1 n)" by simp
with S2_le_S1 have finite_S2: "\n. finite (?S2 n)" by (rule finite_subset)
let ?g = "\(i,j). a i * b j"
let ?f = "\(i,j). norm (a i) * norm (b j)"
have f_nonneg: "\x. 0 \ ?f x" by auto
then have norm_sum_f: "\A. norm (sum ?f A) = sum ?f A"
unfolding real_norm_def
by (simp only: abs_of_nonneg sum_nonneg [rule_format])
have "(\n. (\kk (\k. a k) * (\k. b k)"
by (intro tendsto_mult summable_LIMSEQ summable_norm_cancel [OF a] summable_norm_cancel [OF b])
then have 1: "(\n. sum ?g (?S1 n)) \ (\k. a k) * (\k. b k)"
by (simp only: sum_product sum.Sigma [rule_format] finite_lessThan)
have "(\n. (\kk (\k. norm (a k)) * (\k. norm (b k))"
using a b by (intro tendsto_mult summable_LIMSEQ)
then have "(\n. sum ?f (?S1 n)) \ (\k. norm (a k)) * (\k. norm (b k))"
by (simp only: sum_product sum.Sigma [rule_format] finite_lessThan)
then have "convergent (\n. sum ?f (?S1 n))"
by (rule convergentI)
then have Cauchy: "Cauchy (\n. sum ?f (?S1 n))"
by (rule convergent_Cauchy)
have "Zfun (\n. sum ?f (?S1 n - ?S2 n)) sequentially"
proof (rule ZfunI, simp only: eventually_sequentially norm_sum_f)
fix r :: real
assume r: "0 < r"
from CauchyD [OF Cauchy r] obtain N
where "\m\N. \n\N. norm (sum ?f (?S1 m) - sum ?f (?S1 n)) < r" ..
then have "\m n. N \ n \ n \ m \ norm (sum ?f (?S1 m - ?S1 n)) < r"
by (simp only: sum_diff finite_S1 S1_mono)
then have N: "\m n. N \ n \ n \ m \ sum ?f (?S1 m - ?S1 n) < r"
by (simp only: norm_sum_f)
show "\N. \n\N. sum ?f (?S1 n - ?S2 n) < r"
proof (intro exI allI impI)
fix n
assume "2 * N \ n"
then have n: "N \ n div 2" by simp
have "sum ?f (?S1 n - ?S2 n) \ sum ?f (?S1 n - ?S1 (n div 2))"
by (intro sum_mono2 finite_Diff finite_S1 f_nonneg Diff_mono subset_refl S1_le_S2)
also have "\ < r"
using n div_le_dividend by (rule N)
finally show "sum ?f (?S1 n - ?S2 n) < r" .
qed
qed
then have "Zfun (\n. sum ?g (?S1 n - ?S2 n)) sequentially"
apply (rule Zfun_le [rule_format])
apply (simp only: norm_sum_f)
apply (rule order_trans [OF norm_sum sum_mono])
apply (auto simp add: norm_mult_ineq)
done
then have 2: "(\n. sum ?g (?S1 n) - sum ?g (?S2 n)) \ 0"
unfolding tendsto_Zfun_iff diff_0_right
by (simp only: sum_diff finite_S1 S2_le_S1)
with 1 have "(\n. sum ?g (?S2 n)) \ (\k. a k) * (\k. b k)"
by (rule Lim_transform2)
then show ?thesis
by (simp only: sums_def sum.triangle_reindex)
qed
lemma Cauchy_product:
fixes a b :: "nat \ 'a::{real_normed_algebra,banach}"
assumes "summable (\k. norm (a k))"
and "summable (\k. norm (b k))"
shows "(\k. a k) * (\k. b k) = (\k. \i\k. a i * b (k - i))"
using assms by (rule Cauchy_product_sums [THEN sums_unique])
lemma summable_Cauchy_product:
fixes a b :: "nat \ 'a::{real_normed_algebra,banach}"
assumes "summable (\k. norm (a k))"
and "summable (\k. norm (b k))"
shows "summable (\k. \i\k. a i * b (k - i))"
using Cauchy_product_sums[OF assms] by (simp add: sums_iff)
subsection \<open>Series on \<^typ>\<open>real\<close>s\<close>
lemma summable_norm_comparison_test:
"\N. \n\N. norm (f n) \ g n \ summable g \ summable (\n. norm (f n))"
by (rule summable_comparison_test) auto
lemma summable_rabs_comparison_test: "\N. \n\N. \f n\ \ g n \ summable g \ summable (\n. \f n\)"
for f :: "nat \ real"
by (rule summable_comparison_test) auto
lemma summable_rabs_cancel: "summable (\n. \f n\) \ summable f"
for f :: "nat \ real"
by (rule summable_norm_cancel) simp
lemma summable_rabs: "summable (\n. \f n\) \ \suminf f\ \ (\n. \f n\)"
for f :: "nat \ real"
by (fold real_norm_def) (rule summable_norm)
lemma summable_zero_power [simp]: "summable (\n. 0 ^ n :: 'a::{comm_ring_1,topological_space})"
proof -
have "(\n. 0 ^ n :: 'a) = (\n. if n = 0 then 0^0 else 0)"
by (intro ext) (simp add: zero_power)
moreover have "summable \" by simp
ultimately show ?thesis by simp
qed
lemma summable_zero_power' [simp]: "summable (\n. f n * 0 ^ n :: 'a::{ring_1,topological_space})"
proof -
have "(\n. f n * 0 ^ n :: 'a) = (\n. if n = 0 then f 0 * 0^0 else 0)"
by (intro ext) (simp add: zero_power)
moreover have "summable \" by simp
ultimately show ?thesis by simp
qed
lemma summable_power_series:
fixes z :: real
assumes le_1: "\i. f i \ 1"
and nonneg: "\i. 0 \ f i"
and z: "0 \ z" "z < 1"
shows "summable (\i. f i * z^i)"
proof (rule summable_comparison_test[OF _ summable_geometric])
show "norm z < 1"
using z by (auto simp: less_imp_le)
show "\n. \N. \na\N. norm (f na * z ^ na) \ z ^ na"
using z
by (auto intro!: exI[of _ 0] mult_left_le_one_le simp: abs_mult nonneg power_abs less_imp_le le_1)
qed
lemma summable_0_powser: "summable (\n. f n * 0 ^ n :: 'a::real_normed_div_algebra)"
proof -
have A: "(\n. f n * 0 ^ n) = (\n. if n = 0 then f n else 0)"
by (intro ext) auto
then show ?thesis
by (subst A) simp_all
qed
lemma summable_powser_split_head:
"summable (\n. f (Suc n) * z ^ n :: 'a::real_normed_div_algebra) = summable (\n. f n * z ^ n)"
proof -
have "summable (\n. f (Suc n) * z ^ n) \ summable (\n. f (Suc n) * z ^ Suc n)"
(is "?lhs \ ?rhs")
proof
show ?rhs if ?lhs
using summable_mult2[OF that, of z]
by (simp add: power_commutes algebra_simps)
show ?lhs if ?rhs
using summable_mult2[OF that, of "inverse z"]
by (cases "z \ 0", subst (asm) power_Suc2) (simp_all add: algebra_simps)
qed
also have "\ \ summable (\n. f n * z ^ n)" by (rule summable_Suc_iff)
finally show ?thesis .
qed
lemma summable_powser_ignore_initial_segment:
fixes f :: "nat \ 'a :: real_normed_div_algebra"
shows "summable (\n. f (n + m) * z ^ n) \ summable (\n. f n * z ^ n)"
proof (induction m)
case (Suc m)
have "summable (\n. f (n + Suc m) * z ^ n) = summable (\n. f (Suc n + m) * z ^ n)"
by simp
also have "\ = summable (\n. f (n + m) * z ^ n)"
by (rule summable_powser_split_head)
also have "\ = summable (\n. f n * z ^ n)"
by (rule Suc.IH)
finally show ?case .
qed simp_all
lemma powser_split_head:
fixes f :: "nat \ 'a::{real_normed_div_algebra,banach}"
assumes "summable (\n. f n * z ^ n)"
shows "suminf (\n. f n * z ^ n) = f 0 + suminf (\n. f (Suc n) * z ^ n) * z"
and "suminf (\n. f (Suc n) * z ^ n) * z = suminf (\n. f n * z ^ n) - f 0"
and "summable (\n. f (Suc n) * z ^ n)"
proof -
from assms show "summable (\n. f (Suc n) * z ^ n)"
by (subst summable_powser_split_head)
from suminf_mult2[OF this, of z]
have "(\n. f (Suc n) * z ^ n) * z = (\n. f (Suc n) * z ^ Suc n)"
by (simp add: power_commutes algebra_simps)
also from assms have "\ = suminf (\n. f n * z ^ n) - f 0"
by (subst suminf_split_head) simp_all
finally show "suminf (\n. f n * z ^ n) = f 0 + suminf (\n. f (Suc n) * z ^ n) * z"
by simp
then show "suminf (\n. f (Suc n) * z ^ n) * z = suminf (\n. f n * z ^ n) - f 0"
by simp
qed
lemma summable_partial_sum_bound:
fixes f :: "nat \ 'a :: banach"
and e :: real
assumes summable: "summable f"
and e: "e > 0"
obtains N where "\m n. m \ N \ norm (\k=m..n. f k) < e"
proof -
from summable have "Cauchy (\n. \k
by (simp add: Cauchy_convergent_iff summable_iff_convergent)
from CauchyD [OF this e] obtain N
where N: "\m n. m \ N \ n \ N \ norm ((\kk
by blast
have "norm (\k=m..n. f k) < e" if m: "m \ N" for m n
proof (cases "n \ m")
case True
with m have "norm ((\kk
by (intro N) simp_all
also from True have "(\kkk=m..n. f k)"
by (subst sum_diff [symmetric]) (simp_all add: sum.last_plus)
finally show ?thesis .
next
case False
with e show ?thesis by simp_all
qed
then show ?thesis by (rule that)
qed
lemma powser_sums_if:
"(\n. (if n = m then (1 :: 'a::{ring_1,topological_space}) else 0) * z^n) sums z^m"
proof -
have "(\n. (if n = m then 1 else 0) * z^n) = (\n. if n = m then z^n else 0)"
by (intro ext) auto
then show ?thesis
by (simp add: sums_single)
qed
lemma
fixes f :: "nat \ real"
assumes "summable f"
and "inj g"
and pos: "\x. 0 \ f x"
shows summable_reindex: "summable (f \ g)"
and suminf_reindex_mono: "suminf (f \ g) \ suminf f"
and suminf_reindex: "(\x. x \ range g \ f x = 0) \ suminf (f \ g) = suminf f"
proof -
from \<open>inj g\<close> have [simp]: "\<And>A. inj_on g A"
by (rule subset_inj_on) simp
have smaller: "\n. (\i g) i) \ suminf f"
proof
fix n
have "\ n' \ (g ` {..
by (metis Max_ge finite_imageI finite_lessThan not_le not_less_eq)
then obtain m where n: "\n'. n' < n \ g n' < m"
by blast
have "(\i
by (simp add: sum.reindex)
also have "\ \ (\i
by (rule sum_mono2) (auto simp add: pos n[rule_format])
also have "\ \ suminf f"
using \<open>summable f\<close>
by (rule sum_le_suminf) (simp_all add: pos)
finally show "(\i g) i) \ suminf f"
by simp
qed
have "incseq (\n. \i g) i)"
by (rule incseq_SucI) (auto simp add: pos)
then obtain L where L: "(\ n. \i g) i) \ L"
using smaller by(rule incseq_convergent)
then have "(f \ g) sums L"
by (simp add: sums_def)
then show "summable (f \ g)"
by (auto simp add: sums_iff)
then have "(\n. \i g) i) \ suminf (f \ g)"
by (rule summable_LIMSEQ)
then show le: "suminf (f \ g) \ suminf f"
by(rule LIMSEQ_le_const2)(blast intro: smaller[rule_format])
assume f: "\x. x \ range g \ f x = 0"
from \<open>summable f\<close> have "suminf f \<le> suminf (f \<circ> g)"
proof (rule suminf_le_const)
fix n
have "\ n' \ (g -` {..
by(auto intro: Max_ge simp add: finite_vimageI less_Suc_eq_le)
then obtain m where n: "\n'. g n' < n \ n' < m"
by blast
have "(\ii\{.. range g. f i)"
using f by(auto intro: sum.mono_neutral_cong_right)
also have "\ = (\i\g -` {.. g) i)"
by (rule sum.reindex_cong[where l=g])(auto)
also have "\ \ (\i g) i)"
by (rule sum_mono2)(auto simp add: pos n)
also have "\ \ suminf (f \ g)"
using \<open>summable (f \<circ> g)\<close> by (rule sum_le_suminf) (simp_all add: pos)
finally show "sum f {.. suminf (f \ g)" .
qed
with le show "suminf (f \ g) = suminf f"
by (rule antisym)
qed
lemma sums_mono_reindex:
assumes subseq: "strict_mono g"
and zero: "\n. n \ range g \ f n = 0"
shows "(\n. f (g n)) sums c \ f sums c"
unfolding sums_def
proof
assume lim: "(\n. \k c"
have "(\n. \kn. \k
proof
fix n :: nat
from subseq have "(\kk\g`{.. |
