# Power Series Convergence

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## The Sum May Not Converge

Our formula for the power series is

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

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

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

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

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

can only have three possibilities:

1. The series only converges when x = c
2. The series only converges when ${\displaystyle |x-c|, where R is some constant
3. 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.

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