Power Series

What they is...

The first thing that separates power series from what we have seen with sequences and series is the presence of a new variable \(x\) (or something like it). A power series has the general form of:

\[ \begin{align*} \sum_{n=0}^\infty a_n (x-c)^n = a_1 + a_2 (x-c)^1 + a_2 (x-c)^2 + ... + a_n (x-c)^n + ... \end{align*} \]

We say that the above is a power series in \(x\) centered at \(c\).

The presence of \(x\) here is what makes a power series a function and you can see that is has this sort of neverending polynomial format as $ n \to \infty $.

When we discuss convegence of power series we will find that the series may only converge for certain values of x and for other values the series would diverge.

Interval of Convergence

This defines the values of x for which the power series will converge. The bounds themselves need a convergence test to indicate if they will converge, but all points in between will converge for the given interval.

Radius of Convergence

The radius defines the bound within which the power series will converge: $ | x-a | < R $