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Lesson PreviewLesson 14 of 15 in this unit

10.14 Finding Taylor or Maclaurin Series for a Function

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 Taylor series is the infinite-series version of a Taylor polynomial. Instead of stopping after finitely many terms, you continue forever:
∑n=0∞f(n)(a)n!(x−a)n\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n
When a=0, this is a Maclaurin series. These series represent functions locally as infinite polynomials, and for many important functions, the series converges to the function on a whole interval. In BC calculus, you often find these by recognizing a known series or by using derivatives at the center.

You should know the major standard Maclaurin series and be able to build others from them. The most important ones are for ex,sin⁡x,cos⁡x,and11−x{e^x, \sin x, \cos x, and \frac{1}{1-x}}. Once you know those, you can often substitute expressions, multiply by constants, or shift variables to create new series. This is one of the most powerful parts of Unit 10 because it turns difficult functions into algebraic series you can manipulate term by term.

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 FunctionThis lesson
  15. 10.15Representing Functions as Power SeriesPreview lesson

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