Power Series Convergence

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The Sum May Not Converge

Our formula for the power series is

∑k=0∞ak(x−c)k{\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

∑k=1∞axk−1=a+ax+ax2+ax3+…, a≠0{\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

∑k=0∞ak(x−c)k,{\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 |x−c|<R{\displaystyle |x-c|<R}, 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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