Question

Check whether the sum converges or diverges:
$\boxed{\sum_{n = 1}^ \infty \frac{1}{ \sqrt{n} (n + 1) + n \sqrt{n + 1} }}$
​

Answer 1

$\huge \sf \pink{Converges}$

Step by step

$\displaystyle \color{red} \tt\sum_{n=1}^{\infty} \frac{1}{\sqrt{n} \left(n + 1\right) + n \sqrt{n + 1}}$

Rewrite

$\displaystyle \tt\color{red}{}=\color{red}{\left(\sum_{n=1}^{\infty} \frac{1}{n^{\frac{3}{2}} + \sqrt{n} + n \sqrt{n + 1}}\right)}$

Can't find the exact value.

By the limit comparison test, the series is convergent.

$\tiny\displaystyle \tt\color{red}{\left(\sum_{n=1}^{\infty} \frac{1}{n^{\frac{3}{2}} + \sqrt{n} + n \sqrt{n + 1}}\right)} \approx \color{red}{\left(0.92946543841414\right)}$

Hence,

$\tiny \displaystyle \tt \red{\sum_{n=1}^{\infty} \frac{1}{\sqrt{n} \left(n + 1\right) + n \sqrt{n + 1}} \approx 0.92946543841414}$