$\sum_n \frac {1 } {\sqrt { n}}-\sqrt{\ln{ \frac{n+1 }{ n}} }$ series convergence I need a hint on how to prove whether $\sum_n \frac {1 } {\sqrt { n}}-\sqrt{\ln{ \frac{n+1 }{ n}} }$ is convergent. Probably directly using any of the criteria for series of positive terms would not work here as I have tried this before.

You have that $$\ln\left(\frac{n+1}{n}\right)=\frac{1}{n}-\frac{1}{2n^2}+o\left(\frac{1}{n^2}\right).$$
Therefore $$\sqrt{\ln\left(\frac{n+1}{n}\right)}=\frac{1}{\sqrt n}\sqrt{1-\frac{1}{2n}+o\left(\frac{1}{n}\right)}=\sqrt{\frac{1}{n}}\left(1-\frac{1}{4n}+o\left(\frac{1}{n}\right)\right),$$
and thus $$\frac{1}{\sqrt n}-\sqrt{\ln\left(\frac{n+1}{n}\right)}=\mathcal O\left(\frac{1}{n^{3/2}}\right).$$
I let you conclude.
