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$\sum a_n$ 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.

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  • $\begingroup$ X,T,Chen. Absolute convergence of$ \sum a_n$ and $\sum b_n $? $\endgroup$ – Peter Szilas Apr 15 at 6:08
  • $\begingroup$ @PeterSzilas nope. Just conditionally converge $\endgroup$ – X.T Chen Apr 15 at 6:09
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This can be solved by Cesaro Theorem of Cauchy product.here

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