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Infinite Series
Copyright © Cengage Learning. All rights reserved.

2. 9.9 Representing Functions
by Power Series

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

4. Geometric Power Series

5. 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)

6. 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]

7. 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)

8. 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.

9. Fun Review: Do the division:

10. 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

11. 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.

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

13. 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).

14. 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.

15. 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).

17. 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

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

19. Example 4:
find:
Given:

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

21. Example 5:

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

23. p

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

25. 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

26. Operations with Power Series

27. Operations with Power Series

28. 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.

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

30. 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).

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