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To put it simply: does the series $\sum_{n=1}^{\infty} \frac{n(-1)^{n}}{(2n+1)} = -\frac{1}{3} + \frac{2}{5} - \frac{3}{7} + \cdots$ converge?

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    The terms do not converge to zero, so it does not converge. If you take the terms in pairs, it does converge, but so does $\sum (-1)^n$.2012-11-14

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Hint: Note that the absolute value of the terms does not go to zero, so it fails the alternating series test. Since the absolute value of the terms is greater than $\frac 14$, what limit and $N$ would you select if I give you $\epsilon=\frac 18?$

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    Thank you for your answers. I think I get the idea now.2012-11-14