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Power series

Robert Eisele

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 for 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 \(x\in\left(-\frac{1}{2}, \frac{1}{2}\right)\).