Finding Values for x for which $\sum_{n=1}^\infty \frac{x^n}{3^n}$ converges

Question

My question is to find the values of x for which $\sum_{n=1}^\infty \frac{x^n}{3^n}$ converges and to also find the sum of the series for those values of x.

I was going to use the ratio test, however I technically haven't covered this so I am not allowed to use it. Is there any other way apart from using trial and error to find our values?

Thank you!

Answer 1

This is a geometric series with ratio $\frac x3$. The radius of convergence of a geometric series is $1$, so this series converges iff $\bigl|\frac x3\bigr|<1$, i.e. $|x|<3$.

As to the sum, remember that $$\sum_{n=0}^\infty u^n=\frac1{1-u},\;\text{ so }\quad\sum_{n=1}^{\infty} u^n=u\sum_{n=0}^{\infty} u^n=\frac u{1-u}.$$

Answer 2

Hint: $\left|\frac{x}{3}\right|<1$

---

You series is a geometric series, write it as $\sum_{n=1}^{\infty}{\left(\frac{x}{3}\right)^n}$. Recall that the series $\sum_{n=1}^{\infty}{r^n}$ converges iff $|r|<1$