Science:Infinite Series Module/Units/Unit 3/3.1 Power Series/3.1.04 Power Series Convergence

The Sum May Not Converge

Our formula for the power series is

$\sum _{k=0}^{\infty }a_{k}(x-c)^{k}$

For certain values of x and a_k, a power series can be infinite. Let's go back to the example we introduced earlier in this lesson

$\sum _{k=1}^{\infty }ax^{k-1}=a+ax+ax^{2}+ax^{3}+\ldots ,\ a\neq 0$

The sum of this series that tells us that the series only converges when |x| < 1 (by the divergence test). But in more general power series, there are three distinct possibilities that we can encounter.

Three Possibilities for Convergence

Theorem: Only Three Convergence Results are Possible
A general power series series $\sum _{k=0}^{\infty }a_{k}(x-c)^{k},$ can only have three possibilities: The series only converges when x = c The series only converges when $|x-c|<R$, where R is some constant The series converges for any real value of x

The constant R, if it exists, is called the radius of convergence. The interval of convergence of a power series, is the interval over which the series converges.