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Series converges implies $\lim{n a_n} = 0$

Someone can help me? If $(a_n)$ is a decreasing sequence and $\sum a_n$ converges. Then $\lim {(n.a_n)} = 0$.

I don't have idea how to solve this.

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marked as duplicate by Martin Sleziak, Thomas, rschwieb, Noah Snyder, Arkamis Oct 22 '12 at 18:27

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1 Answer

By the Cauchy condensation test

$$\sum 2^m a_{2^m} < \infty$$

thus

$$\lim_n 2^m a_{2^m} =0$$

Now, for each $n$ chose some $m$ so that $2^m \leq n < 2^{m+1}$ and use

$$a_{2^m} \geq a_n \geq a_{2^{m+1}}$$

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