Difficult limit of an infinite sum

Question

I need a help. There is a limit of an infinite sum I can't calculate. Here it is:

$$\lim_{n\rightarrow\infty}\sum_{j=1}^{n}\frac{(b-1)n}{(j(b-1)+n)^2}=???,\quad\text{($b$=const)}.$$

I haven't got any ideas of how to solve this. But I know that the answer should be $1-\frac{1}{b}$ (it is an area under $y=x^{-2}$ from $1$ to $b$, $b>1$). Hope it will help you.

Answer 1

Hint. Note that $$\sum_{j=1}^{n}\frac{(b-1)n}{(j(b-1)+n)^2}=\frac{(b-1)}{n}\sum_{j=1}^{n}\frac{1}{(\frac{j}{n}(b-1)+1)^2}$$ Now the keyword is Riemann sum. See also Calculate an integral with Riemann sum