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Heyy,

Can we prove or disprove the following

$$ \Sigma a_n \text{ is convergent} \Rightarrow \Sigma a_n^2 \text{ is convergent} $$

Since the statement cannot be proven without knowing whether the series is positive, is there a proper counter example

Thank You

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Try with

$$a_n:=\frac{(-1)^n}{\sqrt n}$$

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  • $\begingroup$ I posted the comment before your edit ;) But now we have two duplicate answers... $\endgroup$ – AlexR May 22 '15 at 16:04
  • $\begingroup$ according to wolframalpha this series is divergent $\endgroup$ – ManZzup May 22 '15 at 16:06
  • $\begingroup$ @Man Then either you mistyped the series or else WA is high...again. The series is convergent, of course. $\endgroup$ – Timbuc May 22 '15 at 16:07
  • $\begingroup$ @G.Sassatelli No, I think you're confusing things... $\endgroup$ – Timbuc May 22 '15 at 16:07
  • $\begingroup$ @Timbuc can u suggest me a test to check if this is convergent? $\endgroup$ – ManZzup May 22 '15 at 16:09
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This is false. Consider $\sum \frac{(-1)^n}{\sqrt{n}}$.

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