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.

We may take the series $\sum_{n\geq 1}\left[\frac{1}{n}-\log\left(1+\frac{1}{n}\right)\right]$ as an equivalent  definition of the Euler-Mascheroni constant $\gamma=\lim_{n\to +\infty}\left(H_n-\log n\right)$, where $H_n=\frac{1}{1}+\frac{1}{2}+\ldots+\frac{1}{n}$ is the $n$-th harmonic number.
Over the interval $(0,1)$ we have that $\frac{x-\log(1+x)}{x^2}$ is a decreasing function, going from $\frac{1}{2}$ to $1-\log(2)$. 
In particular $\gamma$ is bounded between $(1-\log 2)\frac{\pi^2}{6}$ and $\frac{\pi^2}{12}$. More accurate approximations can be derived from creative telescoping, the integral representation
$$ \gamma=\int_{0}^{1}-\log(-\log x)\,dx$$
or the Shafer-Fink inequality. Actually $\gamma$ is pretty close to $\frac{1}{\sqrt{3}}$. The irrationality of $\gamma$ is a long-standing open problem.
