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power series
A power series is a series of the form
$\sum_{{k=0}}^{{\infty}}a_{k}(xx_{0})^{k},$ 
with $a_{k},x_{0}\in\mathbb{R}$ or $\in\mathbb{C}$. The $a_{k}$ are called the coefficients and $x_{0}$ the center of the power series. $a_{0}$ is called the constant term.
Where it converges the power series defines a function, which can thus be represented by a power series. This is what power series are usually used for. Every power series is convergent at least at $x=x_{0}$ where it converges to $a_{0}$. In addition it is absolutely and uniformly convergent in the region $\{x\midxx_{0}<r\}$, with
$r=\liminf_{{k\to\infty}}\frac{1}{\sqrt[k]{a_{k}}}$ 
It is divergent for every $x$ with $xx_{0}>r$. For $xx_{0}=r$ no general predictions can be made. If $r=\infty$, the power series converges absolutely and uniformly for every real or complex $x.$ The real number $r$ is called the radius of convergence of the power series.
Examples of power series are:

Taylor series, for example:
$e^{x}=\sum_{{k=0}}^{{\infty}}\frac{x^{k}}{k!}.$
Power series have some important properties:

If a power series converges for a $z_{0}\in\mathbb{C}$ then it also converges for all $z\in\mathbb{C}$ with $zx_{0}<z_{0}x_{0}$.

Also, if a power series diverges for some $z_{0}\in\mathbb{C}$ then it diverges for all $z\in\mathbb{C}$ with $zx_{0}>z_{0}x_{0}$.

(Uniqueness) If two power series are equal and their centers are the same, then their coefficients must be equal.

Power series can be termwise differentiated and integrated. These operations keep the radius of convergence.
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emphasis on defined terms by Mathprof ✓
Uniform Convergence by azdbacks4234 ✓
link by pahio ✓
not uniform by scineram