Calculus

Here the coefficient is $c_n = \frac{1}{n \cdot 3^n}$. By the ratio test for radius of convergence: $\frac{1}{R} = \lim_{n \to \infty} \left| \frac{c_{n+1}}{c_n} \right| = \lim_{n \to \infty} \frac{1}{(n+1) \cdot 3^{n+1}} \cdot \frac{n \cdot 3^n}{1}$ $\frac{1}{R} = \lim_{n \to \infty} \frac{n}{3(n+1)} = \frac{1}{3} \lim_{n \to \infty} \frac{1}{1 + 1/n} = \frac{1}{3}$ Therefore, the radius of convergence is: $R = 3$