# How do you determine if the series the converges conditionally, absolutely or diverges given Sigma ((-1)^(n))/(lnn) from [1,oo)?

Dec 9, 2016

The series is convergent.

#### Explanation:

We have that ${a}_{n} = \frac{{\left(- 1\right)}^{n}}{\log \left(n\right)}$ It is an alternating series.

This series is converget if

1) ${\lim}_{n \to \infty} {a}_{n} = 0$
2) $\left\mid {a}_{n + 1} \right\mid < \left\mid {a}_{n} \right\mid$

This series accomplishes both requirements so it is convergent.

Attached a plot of $S \left(n\right) = {\sum}_{k = 2}^{n} \frac{{\left(- 1\right)}^{k}}{\log k}$