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Let $\sum a_n$ be a divergent and $\sum b_n$ be a convergent. I want to find $a_n$ , $b_n$ such that $\sum a_nb_n$ is convergent. Does there exist such $a_n$ and $b_n$? If not, how do I prove that?

Clearly, if both $\sum a_n, \;\;\sum b_n$ are divergent then $\sum a_nb_n$ can be convergent. For example, choose $a_n=b_n=\frac{1}{n}$.

Also, if $\sum a_n, \;\;\sum b_n$ are convergent then $\sum a_nb_n$ can be convergent. For example, choose $a_n=b_n=\frac{1}{n^2}$.

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  • $\begingroup$ Why not choose $a_n=1/n$ and $b_n=1/n^2$ then? $\endgroup$ Commented Apr 29, 2020 at 15:30
  • $\begingroup$ Why not $a_n=1$ and your favourite $b_n$? $\endgroup$ Commented Apr 29, 2020 at 15:36

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Actually you almost find the answer. You consider both $a_{n}=b_{n}=\frac{1}{n}$ and $a_{n}=b_{n}=\frac{1}{n^{2}}$. Maybe you should consider the mixture:) In other words, choose $a_{n}=\frac{1}{n}$ and $b_{n}=\frac{1}{n^{2}}$.

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