Convergence of series with alternating signs

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 $\sum (-1)^n$. – Thomas Andrews Nov 14 '12 at 17:33

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 $\epsilon=\frac 18?$
@jeee: The point is that if the series converges, it needs to get close to something, which we call the limit. If the terms stay large, it bounces around too much to converge. The formal definition says that if I give you an $\epsilon \gt 0$, you can give me an $N$ so that any time I sum at least the first $N$ terms (and keep going as far as I want) it will be within $\epsilon$ of the limit. But if $\epsilon = \frac 18$ and each term is at least $\frac 14$, one more term will take you out of the allowable range. – Ross Millikan Nov 14 '12 at 17:55