Does the series $\sum_{n=1}^\infty (-1)^n\ln(n)$ converge or diverge?

Suppose you have the series: $$\sum_{n=1}^\infty (-1)^n\ln(n)$$ Does it converge or diverge? You cannot apply the alternating series test since $b_n$ is not decreasing. Similarly, you cannot apply the nth term test for divergence since the $\lim_{x \to \infty}(-1)^n$ is either $1$ or $-1$.

1 Answer

HINT

Simply check for the necessary condition $a_n\to0$.

Recall indeed that

$$S_n=\sum^n a_k \implies a_n =S_n-S_{n-1}$$

then if

$$S_n \to L\implies a_n =S_n-S_{n-1}\to L-L=0$$