Problem $1$

Infinite Series

a$)[20$ points$]$ Determine whether the series

${\displaystyle \underset{k=1}{\overset{}{}}}\frac{(-2{)}^k}{(2k+1)!}$

converge absolutely$,$ converge conditionally or diverge $?$

b$)[30$ points$]$ Determine the interval of convergence of the following series

${\displaystyle \underset{k=1}{\overset{}{}}}\frac{(-1{)}^k(x+1{)}^k}{(2k+1)}$