Nature of infinite series $ \sum\limits_{n\geq 1}\left[\frac{1}{n} - \log(1 + \frac{1}{n})\right] $ $$\sum\limits_{n\geq 1}\left[\frac{1}{n} - \log\left(1 + \frac{1}{n}\right)\right]$$
Is it convergent or divergent?
Wolfram suggests to use comparison test but I can't find an auxiliary series.

By MVT  applied to $$f:x\mapsto x-\ln(1+x)$$ in $[0,1/n]$,
with $$f'(x)=\frac{x}{1+x}\le x\le \frac 1n$$
$$f(\frac 1n)=\frac 1n f'(c)\le \frac{1}{n^2}$$
by comparison test, the series converges.
