Answer
$ a.\quad R=1.\ \quad $Interval of convergence:$\quad -1 \lt x \lt 1$
$b. \quad$ The interval of absolute convergence is $-1 \lt x \lt 1$
$ c.\quad$ There are no values for which the series converges conditionally
Work Step by Step
(See text: "How to Test a Power Series for Convergence".)
$\text{Step 1.} $
Use the Ratio Test to find the interval where the series converges absolutely.
Ordinarily, $|x-a|\lt R\quad $ or $\quad a-R\lt x\lt a+R.$
$\begin{align*}
& \displaystyle \lim_{n\rightarrow\infty}|\frac{u_{n+1}}{u_{n}}|&\lt 1 \\
& \displaystyle \lim_{n\rightarrow\infty}|\frac{x^{n+1}}{x^{n}}|&\lt 1 \\
& \text{This is true when}\\
& |x|\lt 1 & & \\
& -1 \lt x \lt 1 \end{align*}$
Determine the center and radius:
$a -R\lt x\lt a+R$
$0 -1\lt x\lt 0+1,$
$ a=0,\quad$
the radius is $R=1,$
the interval of convergence is $-1 \lt x \lt 1$
$\text{Step 2.} $
If the interval of absolute convergence is finite,
test for convergence or divergence at each endpoint.
When $x=-1$ $\displaystyle \qquad \sum_{n=1}^{\infty}(-1)^{n} \qquad$ is a divergent series;
When $x=1, \displaystyle \qquad \sum_{n=1}^{\infty}1 \qquad$ is a divergent series.
