<s>[INST] 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.
 [/INST] We have that
$$\frac{1}{n} - \log\left(1 + \frac{1}{n}\right)= \frac{1}{n}-\frac{1}{n}+\frac{1}{2n^2}+O\left(\frac1{n^3}\right)=\frac{1}{2n^2}+O\left(\frac1{n^3}\right)$$
therefore the given series converges by limit comparison test with $\sum \frac 1{n^2}$.
As an alternative, since $\log (1+x)\ge x-\frac12 x^2$ we have
$$\frac{1}{n} - \log\left(1 + \frac{1}{n}\right)\le \frac{1}{n}-\frac{1}{n}+\frac{1}{2n^2}=\frac{1}{2n^2}$$
and therefore the given series converges by comparison test with $\sum \frac 1{2n^2}$.
</s>