Infinite Sums which turn out to be Riemann Integrals

Question

I'm looking for examples of infinite series which look hard to evaluate at first, but become very simple when viewed as a Riemann integral. An example would be $$\frac{1}{n+1}+\frac{1}{n+2}+ \ldots +\frac{1}{2n}=\sum_{i=1}^{n}\frac{1}{n+i}\\=\frac{1}{n}\sum_{i=1}^{n}\frac{1}{1+i/n} \to \int_{0}^{1}\frac{1}{1+x}dx=\log 2$$ (I just think they're cute)