Radius of convergence and interval of convergnce

2. Originally Posted by gichfred
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.