# Math Help - Find the radius of convegence and the radius of convergence

1. ## Find the radius of convegence and the radius of convergence

Please check attachment
Radius of convergence and interval of convergnce

2. Originally Posted by gichfred
Please check attachment
Radius of convergence and interval of convergnce
$\sum\limits_{n = 0}^\infty {\frac{3^n x^n}{n!}} .$

${a_n} = \frac{{{3^n}{x^n}}}
{{n!}} \, \Rightarrow \, {a_{n + 1}} = \frac{{{3^{n + 1}}{x^{n + 1}}}}
{{\left( {n + 1} \right)!}} = \frac{{3x{3^n}{x^n}}}
{{\left( {n + 1} \right)n!}} \, \Rightarrow \, \frac{{{a_{n + 1}}}}
{{{a_n}}} = \frac{{3x}}{{n + 1}}.$

$\mathop {\lim }\limits_{x \to \infty } \left| {\frac{{{a_{n + 1}}}}
{{{a_n}}}} \right| = 3\left| x \right|\mathop {\lim }\limits_{x \to \infty } \frac{1}{{n + 1}} = 0 < 1 \, \Rightarrow \, R = \infty \, \wedge \, x \in \left( { - \infty ;\infty } \right).$

Ie this series converges for any x.