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

Consider the series

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

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

The given series is conditionally convergent.

Proof. By the Leibniz rule we know that if \{ a_n \} is a monotonic decreasing sequence with \lim_{n \to \infty} a_n = 0, then the alternating series \sum_{n=1}^{\infty} (-1)^{n-1} a_n converges. In this case we have

    \[ a_n = \frac{1}{\sqrt{n}} \]

is monotonic decreasing with \lim_{n \to \infty} a_n =0. Hence, the alternating series

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

converges. This convergence is conditional since

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

diverges. \qquad \blacksquare