Compute the next limit $\lim_{n\to\infty}\sum_{i=1}^{n}\frac{1}{n}\left( \frac{1}{\frac{i}{n}+1}\right)$

Question

Compute the next limit

$$\lim_{n\to\infty}\sum_{i=1}^{n}\frac{1}{n}\left( \frac{1}{\frac{i}{n}+1}\right)$$

I have this

$$\lim_{n\to\infty}\sum_{i=1}^{n}\frac{1}{n}\left( \frac{1}{\frac{i}{n}+1}\right)=\lim_{n\to\infty}\sum_{i=1}^{n}\left( \frac{1}{\frac{ni}{n}+n}\right)=\lim_{n\to\infty}\sum_{i=1}^{n}\left( \frac{1}{i+n}\right)$$

If I use the integral criteria can I get some thing?

Answer 1

Hint. Consider $$\sum_{i=1}^{n}\left( \frac{1}{\frac{i}{n}+1}\right)\frac{1}{n}$$ as a Riemann sum related to the integral $$\int_0^1\frac{1}{x+1} dx.$$