Power Series
Power series are a special kind of Series that can be written in the form:
$$ \sum_{n=0}^\infty = c_n(x-a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 … $$
The example is called a power series in $(x-a)$
The special part of power series is that the convergence of the series depends on the value of $x$. To determine for which values of $x$ the series converges, we simply apply a Ratio Test to the series and calculate $x$ for
$$ \frac{|x-a|}{R} < 1 $$
In this case, $a$ is the series’ center of convergence while $R$ is its radius of convergence.
In power series, after solving for $x$ we get a range for $x$. Even though the series converges in this range, we need to check manually if the series also converges at the ends of those ranges.