I have the following series as an exercise but for some reason i cannot prove if it converges or not.
I used the integral test. The series is positive and decreasing so we can use it( i will not write the proof here). That's what the theory says.
The series is the following
$\sum\limits_{n=1}^{\infty }\frac{1}{n \sqrt{n+1}}$
The result i calculate is the following
$2\lim\limits_{t\rightarrow \infty }\frac{1}{3}\sqrt{(t+1)^{3}}-\sqrt{t+1}-\frac{1}{3}\sqrt{1+1}+\sqrt{1+1}= $
$2\lim\limits_{t\rightarrow \infty }\frac{1}{3}\sqrt{(t+1)^{3}}-\sqrt{t+1}+\frac{2}{3}\sqrt{2}$
Can someone help?