How to evaluate the following limit + sum: $ \lim_{n \to \infty} \sum_{i=1}^n \dfrac{1}{(b-a)i + an}$

Question

How to evaluate the following limit + sum: $$ \lim_{n \to \infty} \sum_{i=1}^n \dfrac{1}{(b-a)i + an}$$ It seems to tend to $\dfrac{\ln{(\dfrac{b}{a})}}{b-a}$ but I have no clue how to proof that.

EDIT: I would really prefer a method that does not use Riemann-sums.

Answer 1

Hint:

Rewrite the sum as $$\sum_{i=1}^n \dfrac{1}{(b-a)i + an}=\sum_{i=1}^n \frac{1}{(b-a)\frac in + a}\,\frac1n,$$ and check it is a Riemann sum of a certain function