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A convergent series

Let \sum \limits_{n=1}^{\infty} a_n be a convergent series of positive terms. Prove that there exists a strictly increasing sequence b_n which is also unbounded such that the series \sum \limits_{n=1}^{\infty} a_n b_n also converges.

Solution

We set b_n=\frac{1}{\sqrt{\sum \limits_{k=n}^{\infty}a_k}} and we observe that

    \[b_n a_n < 2\left(\sqrt{\sum_{k=n}^{\infty}a_k}- \sqrt{\sum_{k=n+1}^{\infty}a_k}\right)\]

That’s all folks.

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