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10.13 Radius and Interval of Convergence of Power Series

Part of Infinite Sequences and Series in AP® Calculus BC. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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A power series centered at a has the form
∑n=0∞cn(x−a)n\sum_{n=0}^{\infty} c_n(x-a)^n
For each such series, there is a number R called the radius of convergence such that the series converges when ∣x−a∣<R{|x-a|<R} and diverges when ∣x−a∣>R{|x-a|>R}. The full interval of convergence includes all x-values for which the series actually converges, which means you must also test the endpoints separately after finding R.

You usually find the radius of convergence using the ratio test. After that, you solve the resulting inequality for x, which gives the open interval. Then you plug in each endpoint and test those two series individually, because the ratio test does not decide what happens there. This is one of the main reasons Unit 10 builds all those convergence tests first: power series need them at the endpoints.

In this unit

15 lessons

  1. 10.1Defining Convergent and Divergent Infinite SeriesPreview lesson
  2. 10.2Working with Geometric SeriesPreview lesson
  3. 10.3The nth Term Test for DivergencePreview lesson
  4. 10.4Integral Test for ConvergencePreview lesson
  5. 10.5Harmonic Series and p-SeriesPreview lesson
  6. 10.6Comparison Tests for ConvergencePreview lesson
  7. 10.7Alternating Series Test for ConvergencePreview lesson
  8. 10.8Ratio Test for ConvergencePreview lesson
  9. 10.9Determining Absolute or Conditional ConvergencePreview lesson
  10. 10.10Alternating Series Error BoundPreview lesson
  11. 10.11Finding Taylor Polynomial Approximations of FunctionsPreview lesson
  12. 10.12Lagrange Error BoundPreview lesson
  13. 10.13Radius and Interval of Convergence of Power SeriesThis lesson
  14. 10.14Finding Taylor or Maclaurin Series for a FunctionPreview lesson
  15. 10.15Representing Functions as Power SeriesPreview lesson

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