Taylor or Maclaurin Series for a Function (College Board AP® Calculus BC): Study Guide

Taylor or Maclaurin series for a function

What is the Taylor series for a function?

• Let be a function

  • If the function exists at

  • and if the function is infinitely differentiable at

    • i.e. if the derivative exists for all

  • then the Taylor series for about is

• Note that

  • The Taylor series is a power series

  • The Taylor series can be seen as a Taylor polynomial that 'goes on forever'

    • I.e. the nth degree Taylor polynomial is a truncated Taylor series, which stops at a particular value of

    • A Taylor polynomial for is a partial sum of the Taylor series for

What is the relationship between Taylor series and power series?

• A power series about is any series of the form

  • 

    • where the coefficients can be any sequence of real numbers

• A Taylor series about is the series

  • 

    • where the coefficients must have the specific form

• All Taylor series about for the function are power series

  • As a Taylor series is a power series, it may converge

    • at a single point (i.e. at only)

    • over a finite interval of values

    • or for all real numbers

  • On its interval of convergence, a Taylor series is exactly equal to its function

    • It only becomes an approximation if it is truncated at some value of

• Not all power series about are Taylor series for the function

  • If, however, a power series about is found such that it converges to the function on some interval of convergence (i.e. has a positive radius of convergence), then that power series is the Taylor series for about

  • This means that the Taylor series for a function is unique

    • I.e., there is no other power series that will also be exactly equal to the function

What is the Maclaurin series for a function?

• A Maclaurin series for a function is a special case of a Taylor series

• It is the Taylor series for a function about the point

  • I.e. if the function and all its derivatives exist at , then the Maclaurin series for about is

  

What standard power series should I be familiar with?

• You should know the following four Maclaurin series for standard functions

  • These are used as the basis for constructing the Maclaurin series for other functions

  • See the 'Representing Functions as Power Series' study guide

• 

  • This series converges for , i.e.

  • Note that this is a geometric series with common ratio

    • Remember that the geometric series sum is for

    • The power series version can be found by

      • dividing both sides of that by

      • and substituting

• 

  • This series converges for all real numbers, i.e. for

  • Note that this is an alternating series and is in radians

• 

  • This series converges for all real numbers, i.e. for

  • Note that this is an alternating series and is in radians

• 

  • This series converges for all real numbers, i.e.

• These results are summarized in the table below

Do all Taylor series have an infinite number of non-zero terms?

• For some functions, the derivatives all become zero after a certain point

  • In particular, this is true of polynomial functions

• For example, if you try to calculate the Taylor series for about

  • The derivatives are , , , , , etc.

  • When you put that into the Taylor series formula you get

    

    • Expanding gives which simplifies to

    • I.e., the Taylor series for is just

• The Taylor series for a polynomial function is always just equal to the polynomial function

• For more 'interesting' functions, however, the Taylor series will generally go on forever