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* ) = *e*^{–x} about the point *x* = ln 2.

Step 1: *f* ^{(n)}(*x* ) = *e*^{–x} when n is even and *f* ^{(n)}(*x* ) = *e*^{–x} 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* ) = *xe*^{x}.

Step 1: Investigating the first few derivatives of *f* (*x* ) = *xe*^{x} shows that *f* (^{n})(*x* ) = *xe*^{x} +*ne*^{x}.

Step 2: Evaluating *f* (^{n})(*x* ) = *xe*^{x} + *ne*^{x} at *x* = 0 gives *f *(^{n})(0) = *n*.

Step 3: *f*(*x* ) =

### Common MacLaurin Series

#### MacLaurin Series for the Functions *e*^{x}, 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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