Answer
$\left( { - \infty ,\infty } \right)$
Work Step by Step
$$\eqalign{
& \sum\limits_{n = 0}^\infty {\frac{{{x^{5n}}}}{{n!}}} \cr
& {\text{Let }}{u_n} = \frac{{{x^{5n}}}}{{n!}} \cr
& {\text{Using the ratio test}} \cr
& \mathop {\lim }\limits_{n \to \infty } \left| {\frac{{{u_{n + 1}}}}{{{u_n}}}} \right| = \mathop {\lim }\limits_{n \to \infty } \left| {\frac{{{x^{5\left( {n + 1} \right)}}}}{{\left( {n + 1} \right)!}} \cdot \frac{{n!}}{{{x^{5n}}}}} \right| \cr
& {\text{Simplify}}{\text{, recall that }}\left( {n + 1} \right)! = \left( {n + 1} \right)n! \cr
& = \mathop {\lim }\limits_{n \to \infty } \left| {\frac{{{x^{5n}}{x^5}}}{{\left( {n + 1} \right)n!}} \cdot \frac{{n!}}{{{x^{5n}}}}} \right| \cr
& = \mathop {\lim }\limits_{n \to \infty } \left| {\frac{{{x^5}}}{{n + 1}}} \right| \cr
& = {x^5}\mathop {\lim }\limits_{n \to \infty } \left| {\frac{1}{{n + 1}}} \right| \cr
& {\text{Evaluate the limit}} \cr
& = {x^5}\left( 0 \right) \cr
& = 0 \cr
& {\text{By the Ratio Test}},{\text{ If }}\mathop {\lim }\limits_{n \to \infty } \left| {\frac{{{u_{n + 1}}}}{{{u_n}}}} \right| = L < 1,{\text{ then }}\sum\limits_{n = 0}^\infty {{a_n}} {\text{ converges}} \cr
& {\text{absolutely}}{\text{. Therefore}} \cr
& {\text{The series converges for }}\left( { - \infty ,\infty } \right) \cr} $$
