Interval of pointwise convergence of a function series

Question

$$ \sum _{n=0}^{\infty }\:\left(\frac{x}{3}\right)^n\sin\left(\frac{\pi \:x}{6}\right) $$ The problem is (fill blanks):

The interval of pointwise convergence of the series above is the union of the interval (blank) and points: $$ x\left(k\right)=(blank) $$ (write a function $x(k)$ of an integer k such that: $ x\left(0\right)=0 $)

I got that the convergence interval is $-3<x<3$ aka $(-3, 3)$ using the Ratio test, but I do not understand the union with the $x(k)$ function part.

Answer 1

The series also converges if $\sin{\frac{\pi x}{6}}=0$ so can you come up with a function $x(k)$ with $x(0)=0$ where $\sin{\frac{\pi x(k)}{6}}=0$?