## Various series

Investigate the convergence of the series

• #### Variant 1

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

• #### Variant 2

$$\displaystyle \sum_{n=1}^{\infty}\sqrt\frac{n-1}{2n}$$.

• #### Variant 3

$$\displaystyle \sum_{n=1}^{\infty}\frac{n+1}{(n+1)\sqrt{n+1}-1}$$

• #### Variant 4

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

• #### Variant 5

$$\displaystyle \sum_{n=1}^{\infty}\frac1{(n+3)\sqrt{n}}$$.

• #### Variant 6

$$\displaystyle \sum_{n=1}^{\infty}\frac{\sqrt n}{n^3+1}$$.

• #### Variant 7

$$\displaystyle \sum_{n=1}^{\infty}\sqrt[n]{\frac1{1000}}$$.

• #### Variant 8

$$\displaystyle \sum_{n=1}^{\infty}\frac{2n-1}{\sqrt{2^n}}$$

• #### Variant 9

$$\displaystyle \sum_{n=1}^{\infty}\frac1{n\cdot3^n}$$.

• #### Variant 10

$$\displaystyle \sum_{n=1}^{\infty}\frac{n^5}{2^n+3^n}$$.

• #### Variant 11

$$\displaystyle \sum_{n=1}^{\infty}\frac{1!+2!+3!+…+n!}{(2n)!}$$.

• #### Variant 12

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

• #### Variant 13

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

• #### Variant 14

$$\displaystyle \sum_{n=1}^{\infty}\frac{2+\cos n}{n+\ln n}$$

• #### Variant 15

$$\displaystyle \sum_{n=1}^{\infty}\frac{\sin n}{n(n+2)}$$

• #### Variant 16

$$\displaystyle \sum_{n=1}^{\infty}\sin\frac1n$$.

• #### Variant 17

$$\displaystyle \sum_{n=1}^{\infty}\sin\frac1{n^2}$$.

• #### Variant 18

$$\displaystyle \sum_{n=1}^{\infty}e^{\sqrt[-3]n}$$.

• #### Variant 19

$$\displaystyle \sum_{n=1}^{\infty}\frac{\ln n}{n}$$

• #### Variant 20

$$\displaystyle \sum_{n=2}^{\infty}\frac{\sqrt{n^2+1}-n}{\log^2 n}\ .$$