<s>[INST] Sum of series of ratios I need to calculate
$$\lim_{n \to \infty}\frac{1}{n}\left(\frac{1}{1+n} + \frac{1}{2+n} + \frac{1}{3+n}+\cdots+\frac{3n}{4n}\right).$$
It can be written as 
$$\lim_{n \to \infty}\frac{1}{n^2}\left(\frac{1}{\frac{1}{n}+1} + \frac{1}{\frac{2}{n}+1} + \frac{1}{\frac{3}{n}+1}+\cdots+\frac{1}{\frac{4n}{3n}}\right).$$ 
Now I need to write it as $\frac{1}{n}\sum_{?}^{?}{?}$ and ultimately in definite integral form so that I can evalute the sum of this limit.
 [/INST] Hint:
$$
\frac{1}{n}\sum_{k=1}^{mn} \frac{1}{\frac{k}{n} + 1} \approx \int_0^m \frac{1}{x+1}dx
$$
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