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?
Best Answer
$$\frac{\sqrt{n+1}-\sqrt n}{n}=\frac{1}{n(\sqrt{n+1}+\sqrt n)}<n^{-\frac32}$$