Question

Determine whether the following series converge or diverge

Answer 1

An alternating series $\sum\limits_n(-1)^na_n$ converges if $|(-1)^na_n|=|a_n|$ is monotonic and $a_n\to0$ as $n\to\infty$. Here $a_n=\dfrac1{\ln(n+1)}$.

Let $f(x)=\ln(x+1)$. Then $f'(x)=\dfrac1{x+1}$, which is positive for all $x>-1$, so $\ln(x+1)$ is monotonically increasing for $x>-1$. This would mean $\dfrac1{\ln(x+1)}$ must be a monotonically decreasing sequence over the same interval, and so must $a_n$.

Because $a_n$ is monotonically increasing, but will still always be positive, it follows that $a_n\to0$ as $n\to\infty$.

So, $\sum\limits_n(-1)^na_n$ converges.