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Determine the convergence of the series (-1)n / (n1/2 + (-1)n)

Consider the series

    \[ \sum_{n=2}^{\infty} \frac{(-1)^n}{\sqrt{n} + (-1)^n}. \]

Determine whether the series converges or diverges. If it converges, determine whether it converges conditionally or absolutely.

The given series diverges.

Proof. To show this series diverges, we first consider the series

    \[ \sum_{n=2}^{\infty} \left( \frac{(-1)^n}{\sqrt{n}} - \frac{(-1)^n}{\sqrt{n} + (-1)^n} \right) = \sum_{n=2}^{\infty} \frac{1}{n+(-1)^n \sqrt{n}}. \]

This series diverges by the comparison test since

    \begin{align*}  \frac{1}{n+(-1)^n \sqrt{n}} &\geq \frac{1}{n+\sqrt{n}} \\[9pt]  &\geq \frac{1}{2n} \end{align*}

and we know

    \[ \sum_{n=2}^{\infty} \frac{1}{2n} = \frac{1}{2}\sum_{n=2}^{\infty} \frac{1}{n} \]

diverges. Further more the series

    \[ \sum_{n=2}^{\infty} \frac{(-1)^n}{\sqrt{n}} \]

converges by the Leibniz test since it is monotonically decreasing and has limit 0. But then we must have

    \[ \sum_{n=2}^{\infty} \frac{(-1)^n}{\sqrt{n} + (-1)^n} \]

divergent. (Since if it was convergent, then adding it with the convergent series \sum \frac{(-1)^n}{\sqrt{n}} would converge, but we showed this sum diverged.) \qquad \blacksquare

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