Question

Степенные ряды помогите найти R

Answer 1

$sumlimits _{n=1}^infty frac{(n+1)cdot x^{n}}{3^{n}cdot (n+2)} \\R= limlimits _{n to infty} frac{a_{n}}{a_{n+1}}= limlimits _{n to infty}frac{(n+1)}{3^{n}cdot (n+2)} : frac{(n+2)}{3^{n+1}cdot (n+3)} = limlimits _{n to infty} frac{(n+1)cdot 3^{n}cdot 3cdot (n+3)}{3^{n}cdot (n+2)(n+2)}=\\= limlimits _{n to infty} frac{3cdot (n^2+4n+3)}{n^2+4n+4} =3\\R=3$