Alternating Series Archives

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 $. \qquad \blacksquare$

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Tag: Alternating Series

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