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9.9 Representing Functions by Power Series

Objectives Find a geometric power series that represents a function. Construct a power series using series operations.

Geometric Power Series

interval of convergence. Also: If the function given by: has a radius of convergence of R > 0, then it is continuous and differentiable on its interval of convergence. Also: 1) 2)

Find the intervals of convergence for the following: Ex. 1 Find the intervals of convergence for the following: a) b) c) a) Using the Ratio test: R = 1 Checking endpoints, you will find the Interval of Convergence is: [-1, 1]

Checking endpoints, the Interval of Convergence is: (-1, 1) b) Using the Ratio test: R = 1 Checking endpoints, the Interval of Convergence is: [-1, 1) c) Using the Ratio test: R = 1 Checking endpoints, the Interval of Convergence is: (-1, 1)

Geometric Series: In a geometric series, each term is found by multiplying the preceding term by the same number, r. This converges to if , and diverges if . is the interval of convergence.

Fun Review: Do the division:

Geometric Power Series Consider the function given by f (x) = 1/(1 – x). The form of f closely resembles the sum of a geometric series In other words, if you let a = 1 and r = x, a power series representation for 1/(1 – x), centered at 0, is

Geometric Power Series Of course, this series represents f(x) = 1/(1 – x) only on the interval (–1, 1), whereas f is defined for all x ≠ 1, as shown in Figure 9.22. To represent f in another interval, you must develop a different series.

Geometric Power Series For instance, you could find the power series for the same function f(x)= 1/(1-x) centered at –1.

Geometric Power Series For instance, to obtain the power series centered at –1, you could write which implies that So, for you have which converges on the interval (–3, 1).

Example 1 – Finding a Geometric Power Series Centered at 0 Find a power series for centered at 0. Solution: Writing f(x) in the form a/(1 – r) produces which implies that a = 2 and r = –x/2.

Example 1 – Solution So, the power series for f(x) is cont’d So, the power series for f(x) is This power series converges when which implies that the interval of convergence is (–2, 2).

This is a geometric series where r= -x. Once we have a series that we know, we can find a new series by doing the same thing to the left and right hand sides of the equation. Example 3: This is a geometric series where r= -x. To find a series for , multiply both sides by x. We can find new power series by applying Calculus to known power series

Example 4: Using differentiation: Given: find: So: We differentiated term by term. Can you express the derivative as a summation?

Example 4: find: Given:

So, instead, use a definite integral on: Example 5: Using Integration: Given: find: hmm? So, instead, use a definite integral on:

Example 5:

Example 5: Using Integration: Given: find: at x = 1, C = 0, so

p

So the interval of convergence is: (0, 2) Ex. 6 Find a power series for centered at 1. What is the interval of convergence? Converges when: So the interval of convergence is: (0, 2)

Ex. 7 Find a power series for centered at 1. We just found that: So how could we use this to find a series for ? At x = 1, C = 0. So

Operations with Power Series

Operations with Power Series

Example 8 – Adding Two Power Series Find a power series, centered at 0, for by using partial fractions to rewrite f(x) as the sum of two rational functions with linear denominators.

Example 8 – Adding Two Power Series Find a power series, centered at 0, for Solution: Using partial fractions, f(x)= where and

Example 8 – Solution By adding the two geometric power series and cont’d By adding the two geometric power series and you obtain the following power series. The interval of convergence for this power series is (–1, 1).

Homework Section 9.7 Day 1: pg.674 1-15 odd, 19, 21 Section 9.7 Day 2: MMM 216