Power series.

Definition 10.3.1   A power series is a series of the form

$\displaystyle \underset{n=0}{\overset{+\infty }{\sum}} c_n (x-x_0)^n$    

where $ x$ is a real variable, $ x_0$ is a fixed real number called the center.

The sequence of real numbers $ (c_n)$ is called the sequence of coefficients of the series.

Example 10.3.2       

Example 10.3.3   Consider the series $ \underset{n=0}{\overset{+\infty }{\sum}} \frac {x^n}{n!}$ . we use the ratio test (v.s. 3.7) in order to determine the domain of convergence of the series. Here we have:

$\displaystyle \left\vert \frac {u_{n+1}}{u_n} \right\vert = \frac {\frac {x^{n+1}}{(n+1)!}}{\frac {x^n}{n!}} =\frac {\vert x\vert}{n+1}$    

Thus, $ \underset{n \rightarrow + \infty}{\lim} \left\vert \frac {u_{n+1}}{u_n} \right\vert
=\underset{n \rightarrow + \infty}{\lim}\frac {\vert x\vert}{n+1} = 0$ , for any real $ x$ . It follows that the given power series is convergent for any real $ x$ (its sum is $ e^x$ , v.i. 5.2).

Example 10.3.4   Consider the series $ \underset{n=0}{\overset{+\infty }{\sum}} nx^n$ . we use the ratio test (v.s. 3.7) in order to determine the domain of convergence of the series. Here we have:

$\displaystyle \left\vert \frac {u_{n+1}}{u_n} \right\vert = \frac {(n+1)x^{n+1}}{nx^n} =\vert x\vert \frac {n+1}{n}$    

Thus, $ \underset{n \rightarrow + \infty}{\lim} \left\vert \frac {u_{n+1}}{u_n} \right...
...derset{n \rightarrow + \infty}{\lim} \vert x\vert \frac {n+1}{n} = \vert x\vert$ , for any real $ x$ . It follows that the given power series is convergent for any real $ x$ such that $ \vert x\vert<1$ .

Proposition 10.3.5   If the power series $ \underset{n=0}{\overset{+\infty }{\sum}} c_n x^n$ is convergent for $ x=x_0$ , then it is absolutely convergent for any $ x$ such that $ \vert x\vert<\vert x_0\vert$ .

If this power series is divergent for $ x=x_1$ , then it is divergent for any $ x$ such that $ \vert x\vert>\vert x_1$ .

It follows that the domain of convergence of a power series is either the whole of $ \mathbb{R}$ or an interval centered at $ x_0$ , i.e. an interval of the form $ (x_0-R, x_0+R)$ (maybe closed or open-closed), wherevthe number v$ R$ is called the convergence radius. In the first case, the radius of convergence is said to be infinite. .

An algorithm for testing the convergence of a power series:
\fbox{
\begin{minipage}{12cm}
\begin{enumerate}
\item Use the ratio test or the ...
...he series is divergent for $\vert x-x_0\vert>R$.
\end{enumerate}\end{minipage} }

Noah Dana-Picard 2007-12-28