Let a be a positive number. Then $\lim_{n \to \infty}[\frac{1}{a+n}+\frac{1}{2a+n}+\cdots +\frac{1}{na+n}]$ Problem : 
Let $a$ be a positive number. Then $$\lim_{n \to \infty}\left[\frac{1}{a+n}+\frac{1}{2a+n}+\cdots +\frac{1}{na+n}\right]$$
Please suggest how to proceed in such limit problems, will be of great help thanks.

We have
$$\sum_{k=1}^n \dfrac1{ka+n} = \dfrac1n \sum_{k=1}^n \dfrac1{1+a\cdot \dfrac{k}n} \sim \int_0^1 \dfrac{dx}{1+ax} = \dfrac{\log(1+a)}a$$
