Answer
converges for all $x$
Work Step by Step
Given $$\sum_{n=0}^{\infty} \frac{(-5)^{n}}{n !}(x+10)^{n}$$
Since $a_n = \frac{(-5)^{n}}{n !}(x+10)^{n}$ and $a_{n+1} = \frac{(-5)^{n+1}}{(n+1( !}(x+10)^{n+1}$, then
\begin{aligned}
\rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\
&=\lim _{n \rightarrow \infty}\left|\frac{(-5)^{n+1}(x+10)^{n+1}}{(n+1) !} \cdot \frac{n !}{(-5)^{n}(x+10)^{n}}\right|\\
&=\lim _{n \rightarrow \infty}\left|5(x+10) \frac{1}{n}\right|\\
&=\left|5(x+10) \right| \lim _{n \rightarrow \infty}\frac{1}{n}\\
&=0
\end{aligned}
Then the series converges for all $x$
