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 $-1$.

• Recall that if $a_n \not \to 0$ we cannot have convergence. More rigorously $$\sum a_n \text{converges} \implies a_n \to 0$$ thus if $a_n \not \to 0$ it cannot converge. – gimusi Mar 21 '18 at 6:40
• Do you mean if $b_n = \ln(n)$ and $\lim_{x \to \infty} b_n$ goes to $0$? You cannot take the limit of $a_n = (-1)^n * \ln(n)$. – Art Mar 21 '18 at 6:44
• The relevant fact is that $a_n \not \to 0$. – gimusi Mar 21 '18 at 6:52
• Oh ok . I think I understand. It could be $-\infty$ or $+\infty$, but definitely not $0$. – Art Mar 21 '18 at 6:54
• Exactly it must be zero! – gimusi Mar 21 '18 at 6:57

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

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

• Under what test? AST? @gimusi – Art Mar 21 '18 at 6:39
• It is the first basic check to do. Just observe that $$a_n =S_n-S_{n-1}$$ then suppose that $S_n\to L$, what about $a_n$? – gimusi Mar 21 '18 at 6:48