Si los valores de $\underset{n\to \infty }{\text{lím}}{a}_n=0,$ entonces ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ converge.

Si los valores de ${\displaystyle \sum _{n=1}^{\infty }\left|a_n\right|}$ converge, entonces ${\displaystyle \sum _{n=1}^{\infty }{a}_n}$ converge.

$a_{n}n\; =\frac{3+n^2}{1-n}$

$a_n=\frac{\text{ln}\left(n+1\right)}{\sqrt{n+1}}$

$a_n=\frac{\text{ln}\left(\text{cos}n\right)}{n}$

${\displaystyle \sum _{n=1}^{\infty }\text{ln}\left(\frac{n+1}{n}\right)}$

${\displaystyle \sum _{n=1}^{\infty }\frac{e^n}{n\text{!}}}$

${\displaystyle \sum _{n=1}^{\infty }\frac{{\left(\mathrm{–1}\right)}^n}{\sqrt{n}}}$

${\displaystyle \sum _{n=1}^{\infty }\frac{{\left(\mathrm{–1}\right)}^{n}n\text{!}}{n^n}}$

${\displaystyle \sum _{n=1}^{\infty }\text{cos}\left(\pi n\right)e^{\text{−}n}}$

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

Halle $\underset{n\to \infty }{\text{lím}}{x}_n$ si $b>1,$ $b<1,$ y $b=1.$

Si los valores de $b=\frac{3}{4}$ y $x_0=100,$ calcule $S_{10}$ y $\underset{n\to \infty }{\text{lím}}{S}_n$