Approximating Functions with Polynomials

Taylor Polynomials

Taylor polynomials provide a way to approximate smooth functions near a point using polynomials. The nth order Taylor polynomial for a function f centered at a is given by:

• Definition: The nth order Taylor polynomial of f at a is

• Remainder (Error) Term: If f is (n+1) times differentiable, then for some \xi between a and x:

• Bounding the Error: To ensure the approximation is accurate to a desired number of decimal places, bound by a constant M and solve for n such that .

Example: Approximating to 30 decimal places requires finding n such that .

Power Series and Their Properties

Definition and Structure

A power series is an infinite sum of the form:

• where are coefficients and c is the center.

• Power series can represent functions as infinite degree polynomials.

Convergence of Power Series

The convergence of a power series depends on the value of x:

• Interval of Convergence: The set of x values for which the series converges.

• Radius of Convergence (R): The distance from the center c to the boundary of the interval of convergence.

• For , the series converges absolutely; for , it diverges.

Finding Interval and Radius of Convergence

• Use the Ratio Test or Root Test to determine convergence:

• Ratio Test:

• Root Test:

• Set to solve for x.

Example: For , the radius of convergence is 10, and the interval is .

Geometric Series and Power Series Representations

• The geometric series for is a fundamental example.

• Functions can be represented as power series using geometric series expansions.

Example: for .

Operations on Power Series

• Sum and Difference: The sum or difference of two convergent power series is also a convergent power series.

• Composition: If is a power series and is a polynomial, is a power series.

• Multiplication by Powers: Multiplying a power series by shifts the index.

Differentiation and Integration of Power Series

• A power series can be differentiated or integrated term by term within its interval of convergence.

• Theorem: If converges for , then:

• Warning: The radius of convergence remains unchanged, but the interval of convergence may change after differentiation or integration. Always retest endpoints.

Example:

Examples and Applications

• Find the power series representation for using .

• Find the power series for using .

• Find the power series for and centered at 0.

• Given , find the function it represents and its interval of convergence.

Summary Table: Power Series Properties

Additional info: These notes cover topics from Chapter 11 (Power Series) and related applications from Chapter 4 (Taylor Polynomials), as well as convergence tests from Chapter 10 (Sequences and Series).