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Ch 10.1 Wkst AP Calc BC Name:

Taylor and Maclaurin Series (Polynomials) - Building the Polynomials

Power Series

A power series about x = a (or "centered about x = a " ) is a sum of constants times powers of ( x – a ).

[pic]

In general, a power series is a series that contain variables, rather than a series of constants. A power series that is "centered about x = a " means that a power series will converge when x = a.

Taylor Series

Taylor Series (or Taylor Polynomials) are specific forms of a power series that are used to approximate function values about x = a (or "centered about x = a " ). Taylor Series are in the following form.

[pic]

Note the forms of a Taylor Series and a Power Series. The constants in a power series are values that are functions of n . The constants in a Taylor Series are values at x = a from a function, [pic], and/or its derivatives. In other words, the coefficient of nth term is in the form [pic].

Maclaurin Series

Maclaurin Series are specific forms of a Taylor Series that are used to approximate function values about x = 0

(or "centered about x = 0 " where a = 0 ). Maclaurin Series are in the following form.

[pic]

As in the Taylor Series, since a = 0 the coefficient of nth term is in the form [pic].

Find the Taylor Series centered at x = 0 (ie Maclaurin Series) of the following functions.

1. [pic]

2. [pic]

3. [pic]

4. [pic]

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ANSWERS:

1) [pic]

2) [pic]

3) [pic]

4) [pic]

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