# Thread: 206.11.02.67 a. Find the function represented by: b. Find the interval of convergence

1. ## 206.11.02.67 a. Find the function represented by: b. Find the interval of convergence

$\tiny{206.11.02.67}\\$
$\textit{Find the function represented by:} \\ \textit{Find the interval of convergence}$
\begin{align*}\displaystyle
f_{67}(x)&=\sum_{k=0}^{\infty}
\left[\frac{(x^2+1)}{4}\right]^k \\
&=?
\end{align*}
Let $\displaystyle r= \frac{x^2+ 1}{4}$. Then the sum is $\displaystyle \sum_{k=0}^\infty r^k$ which is a geometric series. Use the usual formula for the sum of the geometric series, $\displaystyle \frac{1}{1- r}$.
$\dfrac{x^2+1}{4} < 1 \Longrightarrow -\sqrt{3} < x < \sqrt{3}$