Prove that $\lim_{n\to \infty} \sum_{i=1} ^n\frac{1}{n+i}=\ln2 $

Question

Need help with the following proof. $$\lim_{n\to \infty} \sum_{i=1} ^n\frac{1}{n+i}=\ln2 $$

Its the night before my maths exam. I know its a silly question but I need to prove whether given equality is true or false. With reason.

Answer 1

Like The limit of a sum $\sum_{k=1}^n \frac{n}{n^2+k^2}$,

$$\lim_{n\to\infty}\sum_{r=1}^n\frac1{n+r}=\lim_{n\to\infty}\frac1n\sum_{r=1}^n\frac1{1+r/n}$$

$$=\int_0^1\frac{dx}{1+x}$$