I want to know if the sum $\sum_{n=1}^\infty\frac{\sqrt{n+1}-\sqrt{n}}{n}$ converges or not. I've tried the ratio and root test but they don't fit. So wolframalpha says that the sum converges by the comparison test. So I've tried to find a convergent majorizing sum (I've tried e.g. $\sum\frac1{n^\alpha}$ with $\alpha>1$ etc.) but I can't find one.
does anybody know one?