1 Power Series
1.1 Power Series
In mathematics, a power series (in one variable) is an infinite series of the form \[ \sum_{n=0}^\infty a_{n}(x-c)^n = a_{0} + a_{1}(x-c)+a_{2}(x-c)^2+\dots \] where \(a_{n}\) represents the coefficient of the \(n\)-th term and \(c\) is a constant.
Power series are useful in mathematical analysis where they arise as the [[memory-calculus-taylor-series]] of infinitely differentiable functions. In fact, [[borels-theorem]] implies that every power series is the Taylor Series of some smooth function.
In many situations, \(c\) (the center of the series) is equal to zero, for instance when considering a [[memory-calculus-taylor-series|Maclaurin Series]], in which case the power series takes the simpler form \[ \sum_{n=0}^\infty a_{n}x^n = a_{0}+a_{1}x+a_{2}x^2 +\dots . \] The Geometric Series, [[exponential-function]] and [[trigonometric-functions]] are all expressed as power series.
1.2 Radius of Convergence
A power series \(\sum_{n=0}^{\infty}a_{n}(x-c)^n\) is convergent for some values of the variable \(x\) (which will always include \(x=c\)) and diverge for other values of \(x\) (possibly all other values). If \(c\) is not the only point of convergence, then there is always a number \(r\) with \(0<r\leq \infty\) such that the series converges whenever \(\lvert x-c \rvert<r\) and diverges whenever \(\lvert x-c \rvert>r\). Such a number \(r\) is called the radius of convergence of the power series.
In general, the radius of convergence of a power series is given by \[ r=\liminf_{n \to \infty}{\lvert a_{n} \rvert ^{-1/n}}, \] or equivalently by, \[ r^{-1}=\limsup_{n \to \infty}{\lvert a_{n} \rvert ^{1/n}}. \]
The following relation is also satisfied if the limit exists \[ r^{-1}=\liminf_{n \to \infty}{\left\lvert \frac{a_{n+1}}{a_{n}} \right\rvert }. \]