Exact expression for series

Question

Can the exact expression for the following series be found, given $|x|<1$? Just curious.

$f(x) = \frac{x^2}{17}+\frac{x^3}{3}+\frac{x^4}{3}+\frac{x^5}{3}+ \ldots$

Answer 1

Separate the first term and factor out a common term, and you find that

\begin{align*} f(x) &= \frac{x^2}{17} + \frac{x^3}{3}\left(1 + x + x^2 + \dots\right) \\ &= \frac{x^2}{17} + \frac{x^3}{3} \frac{1}{1 - x} \end{align*}

from the closed form of a geometric series.