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

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The Sum May Not Converge

Our formula for the power series is

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

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

can only have three possibilities:

  1. The series only converges when x = c
  2. The series only converges when , 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.