## Root test

### Task number: 2933

Investigate the convergence of the following series.

#### Variant 1

\(\displaystyle \sum_{n=1}^{\infty}\left(\frac{n+1}{3n+2}\right)^n \).

#### Variant 2

\(\displaystyle \sum_{n=1}^{\infty}\left(\frac{n+1}{n^2+1}\right)^n \).

#### Variant 3

\(\displaystyle \sum\limits_{n=1}^{\infty} \frac{5^n}{n^5}. \)

#### Variant 4

\(\displaystyle \sum_{n=1}^{\infty}\frac{n^2}{2^n}\ \).

#### Variant 5

\(\displaystyle \sum\limits_{n = 1}^{\infty} \frac{\left(n + \sqrt n\right)^n}{(2n^2 + n)^{\frac{n}{2}}}. \)

#### Variant 6

\(\displaystyle \sum_{n=1}^{\infty}\left(\sqrt[n]{n}-1\right)^n \).