Answer is: option4
\( 3 \)Solution:
The series converges absolutely if \[ \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|<1 \]
Let \[ a_n=\frac{x^n}{(n+1)3^n} \]
Then \[ \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| = \lim_{n\to\infty} \left| \frac{x^{n+1}}{(n+2)3^{n+1}} \cdot \frac{(n+1)3^n}{x^n} \right| \]
\[ = \lim_{n\to\infty} \left| \frac{x}{3}\cdot\frac{n+1}{n+2} \right| = \left|\frac{x}{3}\right| \]
For convergence: \[ \left|\frac{x}{3}\right|<1 \]
So, \[ |x|<3 \]
This describes an interval centered at the origin with radius \[ R=3 \]
