Limit of the difference between two harmonic numbers

Question

I am trying to evaluate the following limit:

$$ \lim_{n \rightarrow \infty} H_{kn} - H_n = \lim_{n \rightarrow \infty} \sum_{i = 1}^{(k - 1)n} \frac{1}{n + 1} $$

I know that the answer is $\ln k$, but I have no idea how to even approach this problem. I have tried somehow equating this to the Taylor expansion of $\ln(1 + (k - 1))$, but to no avail.

Answer 1

HINT:

Use the Riemann sum to evaluate

$$H_{kn}-H_n =\sum_{m=1}^{(k-1)n}\frac1{n+m}=\frac1n \sum_{m=1}^{(k-1)n}\frac{1}{1+m/n}$$

Answer 2

Hint Use the fact that $$\lim_{n} H_{kn}-\ln(kn)=\lim_n H_n-\ln(n)=\gamma$$

Answer 3

Use the asymptotics $$H_p=\gamma +\log \left({p}\right)+\frac{1}{2 p}+O\left(\frac{1}{p^2}\right)$$ making $$H_{kn}-H_n=\log \left({k}\right)+\frac{\frac{1}{k}-1}{2 n}+O\left(\frac{1}{n^2}\right)$$