Determine the convergence of the series (-1)^n-1 / n^s

by RoRi

March 10, 2016

Consider the series

$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^s}.$

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

If $s \leq 0$ then the series diverges since $a_n \neq 0$ as $n \to \infty$ .

If $s > 0$ then the series converges by the Leibniz rule since $\frac{1}{n^s}$ is decreasing and $\lim_{n \to \infty} \frac{1}{n^s} = 0$ .

The convergence is absolute if and only if $s >1$ since

$\sum_{n=1}^{\infty} \left| \frac{(-1)^{n-1}}{n^s} \right| = \sum_{n=1}^{\infty} \frac{1}{n^s}$

converges if and only if $s > 1$ (by the integral test, Example #1 on page 398 of Apostol).

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