Is $\left(\sum\limits_{i=1}^n \left(\dfrac{1}{i}\right) - \log n\right)$ convergent or divergent?

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$e$: $\lim\limits_{n \to \infty}\left(1+\frac{1}{n}\right)^n$ = $\lim\limits_{n \to \infty}\sum\limits_{i=0}^n\left(\frac{1}{i!}\right)$

$\lim\limits_{n \to \infty}\left(1+\frac{1}{n}\right)^n$ = $\lim\limits_{n \to \infty}\sum\limits_{i=0}^n\left(\frac{1}{i!}\right)$ since: (...