1 Power series

A power series is simply a sum of terms each of which contains a variable raised to a non-negative integer power. To illustrate:

$x-x^3+x^5-x^7+\⋯$

$1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\⋯$

are examples of power series. In HELM booklet  16.3 we encountered an important example of a power series, the binomial series:

$1+px+\frac{p\left(p-1\right)}{2!}{x}^2+\frac{p\left(p-1\right)\left(p-2\right)}{3!}{x}^3+\⋯$

which, as we have already noted, represents the function ${\left(1+x\right)}^p$ as long as the variable $x$ satisfies $\left|x\right|<1.$

A power series has the general form

$b_0+b_{1}x+b_2{x}^2+\&ctdot;={\sum }_{p=0}^{\infty }{b}_p{x}^p$

where $b_0,b_1,b_2,\&ctdot;$ are constants. Note that, in the summation notation, we have chosen to start the series at $p=0$ . This is to ensure that the power series can include a constant term $b_0$ since $x^0=1$ .

The convergence, or otherwise, of a power series, clearly depends upon the value of $x$ chosen. For example, the power series

$1+\frac{x}{2}+\frac{x^2}{3}+\frac{x^3}{4}+\&ctdot;$

is convergent if $x=-1$ (for then it is the alternating harmonic series) and divergent if $x=+1$ (for then it is the harmonic series).