# Power series

An infinite series with a constant center $$a\in\mathbb{R}$$ and a coefficient sequence $$(b_n)_{n\geq 0}$$ is called a power series

$P(x) = \sum\limits_{n=0}^\infty b_n\cdot(x-a)^n$

In many situations the center of the series $$a=0$$, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form

$P(x) = \sum\limits_{n=0}^\infty b_n\cdot x^n$

The question that now arises is, for which values $$x$$ ​​does the power series $$P(x)$$ converge? Obvious is that if $$x=a$$ then

$\sum\limits_{n=0}^\infty b_n\cdot(x-a)^n = \underbrace{b_0\cdot 0^0}_{b_0} + \underbrace{b_1\cdot 0^1+b_2\cdot 0^2+b_3\cdot 0^3+...}_{0}=b_0$

converges. In all other cases it depends on the sequence $$(b_n)_{n\geq 0}$$.

## Examples

1. $$\sum\limits_{n=0}^\infty \underbrace{1}_{b_n}\cdot (x-\underbrace{0}_a)^n = \sum_{n=0}^\infty x^n$$ converges for all $$|x|<1$$ (geometric series which is in the convergence interval $$x\in(-1, 1)$$).
2. $$\sum\limits_{n=0}^\infty \underbrace{2^n}_{b_n}\cdot (x-\underbrace{0}_a)^n = \sum_{n=0}^\infty (2x)^n$$ converges for all $$|2x|<1$$, or $$|x|<\frac{1}{2}$$, which is in the convergence interval $$x\in\left(-\frac{1}{2}, \frac{1}{2}\right)$$.

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