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

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    Find the radius of convegence and the radius of convergence

    Please check attachment
    Radius of convergence and interval of convergnce
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    Quote Originally Posted by gichfred View Post
    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}}}<br />
{{n!}} \, \Rightarrow  \, {a_{n + 1}} = \frac{{{3^{n + 1}}{x^{n + 1}}}}<br />
{{\left( {n + 1} \right)!}} = \frac{{3x{3^n}{x^n}}}<br />
{{\left( {n + 1} \right)n!}}  \, \Rightarrow  \, \frac{{{a_{n + 1}}}}<br />
{{{a_n}}} = \frac{{3x}}{{n + 1}}.

    \mathop {\lim }\limits_{x \to \infty } \left| {\frac{{{a_{n + 1}}}}<br />
{{{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.
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