Power Series — Question 1

*Question 1 Find the center, radius of convergence, and interval of convergence of $$\sum_{n=0}^{\infty}\frac{x^n}{3^n}.$$ Test $x=-3$ and $x=3$ separately and identify each resulting endpoint series.

*Question 1 – Solution

$$\sum_{n=0}^{\infty}\frac{x^n}{3^n}=\sum_{n=0}^{\infty}\left(\frac{x}{3}\right)^n.$$ It converges exactly when $|x/3|<1$, or $|x|<3$. Thus the center is $0$ and $R=3$.

• At $x=3$, the series is $\sum1$, which diverges.

• At $x=-3$, the series is $\sum(-1)^n$. Its terms do not approach $0$, so it diverges.

The interval is $\boxed{(-3,3)}$. Neither endpoint is included.