# How do you test the alternating series Sigma (-1)^n/lnn from n is [2,oo) for convergence?

Nov 4, 2017

By the alternating series test criteria, the series converges

#### Explanation:

Suppose that we have a series $\sum {a}_{n}$ and either

${a}_{n} = {\left(- 1\right)}^{n} {b}_{n}$ or ${a}_{n} = {\left(- 1\right)}^{n + 1} {b}_{n}$ where ${b}_{n} \ge 0$ for all n.

Then if,

$1$ ${\lim}_{n \to \infty} {b}_{n} = 0$

and,

${b}_{n}$ is a decreasing sequence

the series $\sum {a}_{n}$ is convergent.

Here, we have

${\sum}_{n = 2}^{\infty} {\left(- 1\right)}^{n} / \ln n = {\sum}_{n = 2}^{\infty} {\left(- 1\right)}^{n} / \ln n = {\sum}_{n = 2}^{\infty} {\left(- 1\right)}^{n} \cdot \frac{1}{\ln} n$

${b}_{n} = \frac{1}{\ln} n$

${\lim}_{n \to \infty} {b}_{n} = {\lim}_{n \to \infty} \left(\frac{1}{\ln} n\right) = 0$

So, by the alternating series test criteria, the series converges