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Taylor Series and MacLaurin Series for AP Calculus

By — McGraw-Hill Professional
Updated on Oct 24, 2011

Practice problems for these concepts can be found at: Series Practice Problems for AP Calculus

Taylor Series

Main Concepts: Taylor Series and MacLaurin Series, Common MacLaurin Series

A Taylor polynomial approximates the value of a function f (x ) at the point x = a. If the function and all its derivatives exist at x = a, then on the interval of convergence, the Taylor series converges to f (x ). The MacLaurin series is the name given to a Taylor series centered at x = 0.

Example 1

Find the Taylor polynomial of degree 3 for f (x )= about the point x = 3.

Example 2

A function f (x ) is approximated by the third order Taylor series 1 + 2(x – 1) – (x – 1)2 + (x –1)3 centered at x = 1. Find f ' (1) and f ''' (1).

Step 2:  f '(1) = 2. 1! = 2 and f '''(1) = 1. 3! = 6

Example 3

Example 4

Find the Taylor series for the function f (x ) = ex about the point x = ln 2.

Step 1:  f (n)(x ) = ex when n is even and f (n)(x ) =   ex when n is odd.

Step 2:  Evaluate f (n)(ln 2) = e–ln 2 = when n is even and f (n)(ln 2) = when n is odd.

Example 5

Find the MacLaurin series for the function f (x ) = xex.

Step 1:  Investigating the first few derivatives of f (x ) = xex shows that f (n)(x ) = xex +nex.

Step 2:  Evaluating f (n)(x ) = xex + nex at x = 0 gives f (n)(0) = n.

Step 3:  f(x ) =

Common MacLaurin Series

MacLaurin Series for the Functions ex, sin x, cos x, and

Familiarity with these common MacLaurin series will simplify many problems.

Practice problems for these concepts can be found at: Series Practice Problems for AP Calculus

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