--- a/src/HOL/Series.thy Thu Dec 24 00:07:51 2020 +0100
+++ b/src/HOL/Series.thy Thu Dec 24 13:10:18 2020 +0000
@@ -627,6 +627,29 @@
end
+text \<open>Biconditional versions for constant factors\<close>
+context
+ fixes c :: "'a::real_normed_field"
+begin
+
+lemma summable_cmult_iff [simp]: "summable (\<lambda>n. c * f n) \<longleftrightarrow> c=0 \<or> summable f"
+proof -
+ have "\<lbrakk>summable (\<lambda>n. c * f n); c \<noteq> 0\<rbrakk> \<Longrightarrow> summable f"
+ using summable_mult_D by blast
+ then show ?thesis
+ by (auto simp: summable_mult)
+qed
+
+lemma summable_divide_iff [simp]: "summable (\<lambda>n. f n / c) \<longleftrightarrow> c=0 \<or> summable f"
+proof -
+ have "\<lbrakk>summable (\<lambda>n. f n / c); c \<noteq> 0\<rbrakk> \<Longrightarrow> 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: "(\<lambda>n. (1/2::real)^Suc n) sums 1"
proof -
have 2: "(\<lambda>n. (1/2::real)^n) sums 2"