Back to Infinite Sequences and Series

Lesson PreviewLesson 15 of 15 in this unit

10.15 Representing Functions as 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.

Start free

Once you know a standard power series, you can create new series representations by algebraic manipulation. You may substitute a new expression in place of x, multiply by a constant, differentiate term by term, or integrate term by term within the interval of convergence. This lets you represent many functions as power series even when you do not start from scratch with derivatives. This topic is about using known series flexibly rather than memorizing isolated formulas.

You should especially recognize the geometric-series base:
11−x=∑n=0∞xnfor∣x∣<1\frac{1}{1-x}=\sum_{n=0}^{\infty} x^n for |x|<1
From that one series, you can create many others. For example, replacing x with -x gives 11+x{\frac{1}{1+x}}, and integrating term by term gives a logarithmic series. This is one of the most useful skills in BC calculus because it connects convergence, Taylor series, and algebraic manipulation into one unified method.

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 SeriesPreview lesson
  14. 10.14Finding Taylor or Maclaurin Series for a FunctionPreview lesson
  15. 10.15Representing Functions as Power SeriesThis lesson

Continue the full lesson in Stellar

Create a free account to unlock the remaining sections, checkpoints, course practice, and progress tracking for AP® Calculus BC.

Start free