What is
$\displaystyle \lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k+n}$
??
I guessed that the answer would be $\displaystyle \log(2)$, and simulation confirms it, but how do I prove it?
By Riemann sums: $\displaystyle \sum_{k=1}^n\frac 1{k+n}=\frac 1n\sum_{k=1}^n}\frac 1{\frac kn+1}=\frac 1n\sum_{k=1}^nf\left(\frac kn\right)$, where $\displaystyle f(x)=\frac 1{x+1}$. Since $\displaystyle \int_0^1f(t)dt=\log 2$, you are done.