How do you test the alternating series Sigma (-1)^(n+1)/sqrtn from n is [1,oo) for convergence?

May 16, 2017

A sufficient condition for an alternating series $\sum {\left(- 1\right)}^{n} {a}_{n}$ to converge is that:

(1) $\lim {a}_{n} = 0$

(2) ${a}_{n + 1} \le {a}_{n}$

The series:

${\sum}_{n = 1}^{\infty} {\left(- 1\right)}^{n + 1} / \sqrt{n}$

is then convergent, since:

$\lim \frac{1}{\sqrt{n}} = 0$

and

$\frac{1}{\sqrt{n + 1}} < \frac{1}{\sqrt{n}}$