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Consider a sequence $\{a_n\} \to 0$. Does it imply that the series $s_n = \sum\limits_{k = 0}^{n} a_n^2$ converges ?

for example the sequence $a_n = \frac{1}{n}$, $a_n \to 0$ and $\sum\limits_{k=1}^{n}\frac{1}{k^2}$ converges.

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Did you not try $a_n = \frac{1}{\sqrt{n}}$? There doesn't seem to be much thought put into this question. – Pete L. Clark Jun 11 '11 at 13:03
up vote 3 down vote accepted

The answer is no. Consider $\{\frac{1}{\ln(n)}\} \rightarrow 0$. But the corresponding sum diverges. You even have that $\ln(n)^2 < n$ for $n$ big enough so it will "diverge faster" than the harmonic series.

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