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

Question

$\sum a_n$ 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 '19 at 06:08

score 0 · Accepted Answer · answered Apr 15 '19 at 11:50

0

This can be solved by Cesaro Theorem of Cauchy product.here

answered Apr 15 '19 at 11:50

XT Chen

1,506

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

1 Answers1