TITLE: Compute the limit $\lim_{n\to\infty}\frac{1}{\log n}\sum_{k=1}^n\left(1-\frac{1}{n}\right)^k\frac{1}{k}$
QUESTION [8 upvotes]: Similar to this problem, how can one compute the following limit:
  $$\lim_{n\to\infty}\frac{1}{\log n}\sum_{k=1}^n\left(1-\frac{1}{n}\right)^k\frac{1}{k}\quad ?$$

Note that
$$\log x = \sum_{k=1}^{\infty}\left(1-\frac{1}{x}\right)^k\frac{1}{k},\quad x\ge \frac{1}{2}.$$

REPLY [6 votes]: $$\left ( 1-\frac1{n}\right )^k = e^{k \log{(1-1/n)}}  = e^{-k/n + O(1/n^2)} = 1-\frac{k}{n}+O \left ( \frac1{n^2} \right )$$
So as $n \to \infty$,
$$\begin{align}\frac1{\log{n}} \sum_{k=1}^n \left ( 1-\frac1{n}\right )^k \frac1{k} &= \frac1{\log{n}} \sum_{k=1}^n \left (\frac1{k}- \frac1{n} + O \left ( \frac1{n^2} \right ) \right ) \\ &= \frac{H_n - 1}{\log{n}} +  O \left ( \frac1{n \log{n}} \right ) \\ &= 1 - \frac{1-\gamma}{\log{n}} + O \left ( \frac1{n \log{n}} \right ) \end{align}$$
The limit as $n \to \infty$ is thus $1$.