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=\frac1{\ln(n+1)}`.

Let
`f(x)=\ln(x+1)`. Then
`f'(x)=\frac1{x+1}`, which is positive for all
`x>-1`, so
`\ln(x+1)` is monotonically increasing for
`x>-1`. This would mean
`\frac1{\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.