Convergent series

Study the convergence of the series \sum_{n=2}^{\infty}\frac{(-1)^n}{\sqrt{n}+(-1)^n}



Solution. Rewrite in the form \sum_{n=2}^{\infty}\frac{(-1)^n}{\sqrt{n}+(-1)^n}=\sum_{n=1}^{\infty}\left(\frac{(-1)^{2n}}{\sqrt{2n}+(-1)^{2n}}+\frac{(-1)^{2n+1}}{\sqrt{2n+1}+(-1)^{2n+1}}\right) (essentially grouping the summand with even power of -1 with the consequent one)

Now the general term could be rewritten in the form \frac{-2 - \sqrt{2n} + \sqrt{  1 + 2 n}}{(1 + \sqrt{2n}) (-1 +\sqrt{1 + 2 n})}

These are all negative. The numerator tends to -2 (since - \sqrt{2n} + \sqrt{  1 + 2 n}\longrightarrow 0). The denominator is of order 2n, thus \frac{-2 - \sqrt{2n} + \sqrt{  1 + 2 n}}{(1 + \sqrt{2n}) (-1 +\sqrt{1 + 2 n})}\sim \frac{-1}{n}
Thus the series is convergent if and only if the series \sum_{n=1}^{\infty}\frac{-1}{n} is convergent, but the latter diverges, hence the original series diverges.

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