Answer
a) The radius of convergence is $\infty$ and the interval of convergence is: $R$ ($-\infty \lt x\lt \infty$).
b) Interval of absolute convergence: $R$ ($ -\infty \lt x\lt \infty$).
c) We get no other values of $x$ for which the given series converges conditionally.
Work Step by Step
Apply the Ratio Test to compute the interval where the series converges absolutely.
$|\dfrac{u_{n+1}}{u_{n}}|=\lim\limits_{n \to \infty} \dfrac{(3)^{n+1} x^{n+1}}{(n+1)!} \cdot \dfrac{n!}{(3)^n x^n}=(3) |x| \lim\limits_{n \to \infty} |\dfrac{n!}{(n+1)!}|=(3) |x|\lim\limits_{n \to \infty} \dfrac{n!}{(n+1)n!}=(3) |x| \lim\limits_{n \to \infty} \dfrac{1}{(n+1)}=0 \lt 1 $, for all $x \in R$
Thus, the series is absolutely convergent for all $x \in R$. This means that series is convergent for all $x \in R$.
So, we can conclude from the above discussion that:
a) The radius of convergence is $\infty$ and the interval of convergence is $R$ ($-\infty \lt x\lt \infty$).
b) Interval of absolute convergence: $R$ ($ -\infty \lt x\lt \infty$).
c) We get no other values of $x$ for which the given series converges conditionally.
