# Math Help - Simple limit escaping me

1. ## Simple limit escaping me

What is

$\lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k+n}$

??

I guessed that the answer would be $\log(2)$, and simulation confirms it, but how do I prove it?

2. ## Re: Simple limit escaping me

By Riemann sums: $\sum_{k=1}^n\frac 1{k+n}=\frac 1n\sum_{k=1}^n}\frac 1{\frac kn+1}=\frac 1n\sum_{k=1}^nf\left(\frac kn\right)$, where $f(x)=\frac 1{x+1}$. Since $\int_0^1f(t)dt=\log 2$, you are done.