I was trying to solve some last year exam questions. I found the following problem.

If $\{a_n\}$ is a sequence of real numbers such that $\{n^2a_n\}$ is a convergent sequence then prove that the series $\sum a_n$ is also convergent.

I know that to check the convergence of a series, we first find the sequence of partial sums of the series, and if the sequence of partial sums of the series is convergent then the series is also called convergent.

I am unable to move forward from here in any direction. Kindly help. Any hint of suggestion would be helpful to me. Thanks a lot for the help.


1 Answer 1


$(n^{2}a_n)$ is bounded.If $|n^{2}a_n| \leq M$ then $|a_n| \leq \frac M {n^{2}}$ and $\sum \frac M {n^{2}}$ is convergent. Hence $\sum a_n$ is absolutely convergent.


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