Find all 
such that the series
![\[ \sum_{n=0}^{\infty} \frac{x^n}{3^{n+1}} \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-42f070a4c025d6b52d7f91d9242af30f_l3.png "Rendered by QuickLaTeX.com")
converges and compute the sum.
From the geometric series expansion we have,
![\begin{align*} \sum_{n=0}^{\infty} \frac{x^n}{3^{n+1}} &= \frac{1}{3} \cdot \sum_{n=0}^{\infty} \left( \frac{x}{3} \right)^n \\[9pt] &= \frac{1}{3} \left( \frac{1}{1-\frac{x}{3}} \right) \\[9pt] &= \frac{1}{3-x}. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-33655a0c8a440a70781b60b983416cc6_l3.png "Rendered by QuickLaTeX.com")
This is valid for 
.
Find all such that the series
converges and compute the sum.
From the geometric series expansion we have,
This is valid for .