$\sum_n \frac {1 } {\sqrt { n}}-\sqrt{\ln{ \frac{n+1 }{ n}} }$ series convergence

Question

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.

Answer 1

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.

Answer 2

Note that

$$\left|{1\over\sqrt n}-\sqrt{\ln\left(n+1\over n\right)} \right|=\left|{{1\over n}-\ln\left(1+{1\over n} \right) \over{1\over\sqrt n}+\sqrt{\ln\left(1+{1\over n} \right)}} \right|\le\sqrt n\left|{1\over n}-\ln\left(1+{1\over n} \right) \right|$$

and

$$\left|{1\over n}-\ln\left(1+{1\over n} \right)\right|=\left|\int_0^{1/n}\left(1-{1\over1+x}\right)dx\right|=\int_0^{1/n}{x\over1+x}\,dx\le\int_0^{1/n}x\,dx={1\over2n^2}$$

(where we drop the absolute value once things are obviously nonnegative). It follows that

$$\sum_n\left|{1\over\sqrt n}-\sqrt{\ln\left(n+1\over n\right)} \right|\le\sum_n{1\over2n^{3/2}}$$

so the series is absolutely convergent.