# How to prove this series converges…

Beloved community,

Does the following series converge?

$$\sum_{n=0}^\infty \left(\sqrt[n]{n} - \sqrt[n+1]{n+1}\right)$$

According to Wolfram Alpha, it does by the Comparison Test. However, after thinking about it long and hard, I still haven't found any series to compare it to.

$$\sum_{k=1}^n\left(\sqrt[k]k-\sqrt[k+1]{k+1}\right)=1-\sqrt2+\sqrt2-\sqrt[3]3+\sqrt[3]3-\sqrt[4]4+\ldots+\sqrt[n]n-\sqrt[n+1]{n+1}=$$
$$=1-\sqrt[n+1]{n+1}\xrightarrow[n\to\infty]{}0$$