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# Taylor Series and MacLaurin Series for AP Calculus

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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 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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