src/HOL/Series.thy

   from order_tendstoD(2)[OF this zero_less_one] obtain n where "norm c ^ n < 1"
     by (auto simp: eventually_at_top_linorder)
   then show "norm c < 1" using one_le_power[of "norm c" n]
     by (cases "norm c \<ge> 1") (linarith, simp)
 qed (rule summable_geometric)
+
+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 -