# If $\sum c_n$ is convergent where $c_n = a_1b_n + \cdots +a_nb_1$, then $\sum c_n = \sum a_n\sum b_n$

$$\sum a_n$$ and $$\sum b_n$$ is convergent and $$c_n = a_1b_n + \cdots + a_nb_1$$. Prove that if $$\sum c_n$$ is convergent then $$\sum c_n = \sum a_n \sum b_n$$.

This question has been answered using Abel's Lemma in this post. But I am looking forward to a more basic solution which only uses basic definition and Cauchy's convergence test.

• X,T,Chen. Absolute convergence of$\sum a_n$ and $\sum b_n$? – Peter Szilas Apr 15 at 6:08
• @PeterSzilas nope. Just conditionally converge – XT Chen Apr 15 at 6:09