Power Series Radii and Intervals of Convergence. First some examples Consider the following example...
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Transcript of Power Series Radii and Intervals of Convergence. First some examples Consider the following example...
![Page 1: Power Series Radii and Intervals of Convergence. First some examples Consider the following example series: What does our intuition tell us about the.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551c16fb550346a84f8b56bb/html5/thumbnails/1.jpg)
Power Series
Radii and Intervals of Convergence
![Page 2: Power Series Radii and Intervals of Convergence. First some examples Consider the following example series: What does our intuition tell us about the.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551c16fb550346a84f8b56bb/html5/thumbnails/2.jpg)
First some examples
Consider the following example series:
0
2
!
k
k k
• What does our intuition tell us about the convergence or divergence of this series?
• What test should we use to confirm our intuition?
![Page 3: Power Series Radii and Intervals of Convergence. First some examples Consider the following example series: What does our intuition tell us about the.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551c16fb550346a84f8b56bb/html5/thumbnails/3.jpg)
Power Series
Now we consider a whole family of similar series:
0
2
!
k
k k
0
1
!
k
k k
0
4
!
k
k k
0
12!
k
k k
0
214
!
k
k k
• What about the convergence or divergence of these series?
• What test should we use to confirm our intuition?
We should use the ratio test; furthermore, we can use the similarity between the series to test them all at once.
0 !
k
k
x
k
![Page 4: Power Series Radii and Intervals of Convergence. First some examples Consider the following example series: What does our intuition tell us about the.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551c16fb550346a84f8b56bb/html5/thumbnails/4.jpg)
Power Series
Now we consider a whole family of similar series:
0
2
!
k
k k
0
1
!
k
k k
0
4
!
k
k k
0
12!
k
k k
0
214
!
k
k k
• What about the convergence or divergence of these series?
• What test should we use to confirm our intuition?
But remember that the ratio test applies only to series with positive terms---we will be testing for absolute convergence!
0 !
k
k
x
k
![Page 5: Power Series Radii and Intervals of Convergence. First some examples Consider the following example series: What does our intuition tell us about the.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551c16fb550346a84f8b56bb/html5/thumbnails/5.jpg)
How does it go? We start by setting up the appropriate limit.
1!
lim1 !
k
kk
x k
kx
1
1 !lim
!
k
kk
x
k
x
k
lim
1k
x
k
0
Since the limit is 0 which is less than 1, the ratio test tells us that the series
converges absolutely for all values of x.
0 !
k
k
x
k
Why the absolute values?Why on the x’s and not elsewhere?
The series is an example of a power series.0 !
k
k
x
k
![Page 6: Power Series Radii and Intervals of Convergence. First some examples Consider the following example series: What does our intuition tell us about the.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551c16fb550346a84f8b56bb/html5/thumbnails/6.jpg)
What are Power Series?It’s convenient to think of a power series as an infinite polynomial:
Polynomials:
Power Series:
2 3 4
0
3 3 3 3 1 31
3! 5! 7! 9! 2 1 !
k k
k
x x x x x
k
2 311 ( 1) 3( 1) ( 1)4x x x
2 52 3 12x x x
2 3 4
0
1 2 3 4 5 ( 1) k
k
x x x x k x
![Page 7: Power Series Radii and Intervals of Convergence. First some examples Consider the following example series: What does our intuition tell us about the.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551c16fb550346a84f8b56bb/html5/thumbnails/7.jpg)
In general. . .
Definition: A power series is a (family of) series of the form
00
( ) .nn
n
a x x
In this case, we say that the power series is based at x0 or that it is centered at x0.
What can we say about convergence of power series? A great deal, actually.
![Page 8: Power Series Radii and Intervals of Convergence. First some examples Consider the following example series: What does our intuition tell us about the.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551c16fb550346a84f8b56bb/html5/thumbnails/8.jpg)
Checking for Convergence
I should use the ratio test. It is the test of choice when testing for convergence of power series!
![Page 9: Power Series Radii and Intervals of Convergence. First some examples Consider the following example series: What does our intuition tell us about the.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551c16fb550346a84f8b56bb/html5/thumbnails/9.jpg)
Checking for Convergence
Checking on the convergence of
2 3 4
0
1 2 3 4 5 ( 1) k
k
x x x x k x
We start by setting up the appropriate limit.
x1
( 2)lim
( 1)
k
kk
k x
k x
( 2)
lim( 1)k
k x
k
The ratio test guarantees convergence provided that this limit is less than 1. That is, when |x| < 1.
0 11
![Page 10: Power Series Radii and Intervals of Convergence. First some examples Consider the following example series: What does our intuition tell us about the.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551c16fb550346a84f8b56bb/html5/thumbnails/10.jpg)
Checking for Convergence
Checking on the convergence of
2 3 4
0
1 2 3 4 5 ( 1) k
k
x x x x k x
We start by setting up the appropriate limit.
x1
( 2)lim
( 1)
k
kk
k x
k x
( 2)
lim( 1)k
k x
k
The ratio test is inconclusive if |x| = 1. (x = 1.)We have to test these separately!0 11
![Page 11: Power Series Radii and Intervals of Convergence. First some examples Consider the following example series: What does our intuition tell us about the.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551c16fb550346a84f8b56bb/html5/thumbnails/11.jpg)
Checking for Convergence
Checking on the convergence of
2 3 4
0
1 2 3 4 5 ( 1) k
k
x x x x k x
We start by setting up the appropriate limit.
x1
( 2)lim
( 1)
k
kk
k x
k x
( 2)
lim( 1)k
k x
k
The ratio test is inconclusive if |x| = 1. (x = 1.)We have to test these separately!0 11
![Page 12: Power Series Radii and Intervals of Convergence. First some examples Consider the following example series: What does our intuition tell us about the.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551c16fb550346a84f8b56bb/html5/thumbnails/12.jpg)
One More Example:
Determine where the series
We start by setting up the ratio test limit.
13
2( 1) 1 !lim
3
2 1 !
k
kk
x
k
x
k
13 (2 1)!
lim(2 3)!3
k
kk
x k
kx
0
1 3
2 1converg .
!es
k k
k
x
k
3
lim2 2 2 3k
x
k k
3 1 2 3 4 2 1 2 (2 1)lim
1 2 3 4 2 1 2 (2 1) 2 2 2 3k
x k k k
k k k k k
Since the limit is 0 (which is less than 1), the ratio test says that the series converges absolutely no matter what x is.
0
3
![Page 13: Power Series Radii and Intervals of Convergence. First some examples Consider the following example series: What does our intuition tell us about the.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551c16fb550346a84f8b56bb/html5/thumbnails/13.jpg)
What about the convergence of
We start by setting up the ratio test limit.
13
2( 1) 1 !lim
3
2 1 !
k
kk
x
k
x
k
13 (2 1)!
lim(2 3)!3
k
kk
x k
kx
2 3 4
0
3 3 3 3 1 31 ?
3! 5! 7! 9! 2 1 !
k k
k
x x x x x
k
3
lim2 2 2 3k
x
k k
3 1 2 3 4 2 1 2 (2 1)lim
1 2 3 4 2 1 2 (2 1) 2 2 2 3k
x k k k
k k k k k
Since the limit is 0 (which is less than 1), the ratio test says that the series converges absolutely no matter what x is.
0
3
![Page 14: Power Series Radii and Intervals of Convergence. First some examples Consider the following example series: What does our intuition tell us about the.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551c16fb550346a84f8b56bb/html5/thumbnails/14.jpg)
Now you work out the convergence of
2 3 4
0
3 3 3 3 1 31
3 5 7 9 2 1
k k
k
x x x x x
k
Don’t forget those absolute values!
![Page 15: Power Series Radii and Intervals of Convergence. First some examples Consider the following example series: What does our intuition tell us about the.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551c16fb550346a84f8b56bb/html5/thumbnails/15.jpg)
Now you work out the convergence of
We start by setting up the ratio test limit.
13
2( 1) 1lim
3
2 1
k
kk
x
k
x
k
13 (2 1)
lim(2 3)3
k
kk
x k
kx
2 3 4
0
3 3 3 3 1 31
3 5 7 9 2 1
k k
k
x x x x x
k
What does this tell us?
(2 1)3 lim
(2 3)k
kx
k
3x
• The power series converges absolutely when | x+3| < 1. • The power series diverges when | x+3 | > 1. • The ratio test is inconclusive for x = -4 and x = -2. (Test
these separately… what happens?)
![Page 16: Power Series Radii and Intervals of Convergence. First some examples Consider the following example series: What does our intuition tell us about the.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551c16fb550346a84f8b56bb/html5/thumbnails/16.jpg)
Convergence of Power SeriesWhat patterns can we see? What conclusions can we draw?
When we apply the ratio test, the limit will always be either 0 or some positive number times | x – x0 |. (Actually, it could be , too. What would this mean?)
• If the limit is 0, the ratio test tells us that the power series converges absolutely for all x.
• If the limit is k | x - x0 |, the ratio test tells us that the series converges absolutely when k | x - x0 | < 1. It diverges when k | x - x0 | > 1. It fails to tell us anything if k | x - x0 | = 1.
What does this tell us?
![Page 17: Power Series Radii and Intervals of Convergence. First some examples Consider the following example series: What does our intuition tell us about the.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551c16fb550346a84f8b56bb/html5/thumbnails/17.jpg)
0
1 when | - | the series converges absolutely.x x
k
0
1 when | - | the series diverges.x x
k
0
1 when | - | we don't know.x x
k
Suppose that the limit given by the ratio test is 0| - | .k x x
We need to consider separately the cases when
• k | x - x0 | < 1 (the ratio test guarantees convergence),• k | x - x0 | > 1 (the ratio test guarantees divergence), and • k | x - x0 | = 1 (the ratio test is inconclusive).
This means that . . . Recall that k 0 !
![Page 18: Power Series Radii and Intervals of Convergence. First some examples Consider the following example series: What does our intuition tell us about the.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551c16fb550346a84f8b56bb/html5/thumbnails/18.jpg)
Recap
0
1 when | - | the series converges absolutely.x x
k
0
1 when | - | the series diverges.x x
k
0
1 when | - | we don't know. x x
k
0x 0
1x
k0
1x k
Must test endpoints separately!
![Page 19: Power Series Radii and Intervals of Convergence. First some examples Consider the following example series: What does our intuition tell us about the.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551c16fb550346a84f8b56bb/html5/thumbnails/19.jpg)
ConclusionsTheorem: If we have a power series ,
• It may converge only at x = x0.
00
( )nn
n
a x x
• It may converge for all x.
• It may converge on a finite interval centered at x=x0.
0x 0x R0x R
Radius of conv. is R.
0x
0x
Radius of convergence is 0
Radius of conv. is infinite.
![Page 20: Power Series Radii and Intervals of Convergence. First some examples Consider the following example series: What does our intuition tell us about the.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551c16fb550346a84f8b56bb/html5/thumbnails/20.jpg)
Radius of Convergence vs. Interval of Convergence
The set of x values for which the power series
converges, is called the interval of convergence of the power series. The radius of this interval is called the radius of convergence of the power series.
00
( )nn
n
a x x
0
( 1) k
k
k x
For Example:
0 11
Interval of convergence is (-1,1)Radius of convergence is 1
![Page 21: Power Series Radii and Intervals of Convergence. First some examples Consider the following example series: What does our intuition tell us about the.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551c16fb550346a84f8b56bb/html5/thumbnails/21.jpg)
Radius of Convergence vs. Interval of Convergence
Examples:
0
Interval of convergence is (- , )Radius of convergence is infinite
0 !
k
k
x
k
Interval of convergence is (-4 , -2]Radius of convergence is 1
0
1 3
2 1
k k
k
x
k
3 24
![Page 22: Power Series Radii and Intervals of Convergence. First some examples Consider the following example series: What does our intuition tell us about the.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551c16fb550346a84f8b56bb/html5/thumbnails/22.jpg)
We start by setting up the ratio test limit.
0
2 2
ln( 3)converges.
kk
k
x
k
132 2 lim
14
k
kx
k
112 2
ln( 4)lim
2 2
ln( 3)
kk
kkk
x
k
x
k
112 2 ln( 3)lim
ln( 4)2 2
kk
kkk
x k
kx
ln( 3)2 2 lim
ln( 4)k
kx
k
42 2 lim 2 2
3k
kx x
k
The ratio test guarantees convergence provided that 2|x+2| < 1. That is, if |x+2| < 1/2
Where does the series diverge? Where does the
ratio test fail us?
One More Example:
Determine where the series
![Page 23: Power Series Radii and Intervals of Convergence. First some examples Consider the following example series: What does our intuition tell us about the.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551c16fb550346a84f8b56bb/html5/thumbnails/23.jpg)
What about the convergence of
We start by setting up the ratio test limit.112 2
ln( 4)lim
2 2
ln( 3)
kk
kkk
x
k
x
k
112 2 ln( 3)lim
ln( 4)2 2
kk
kkk
x k
kx
ln( 3)2 2 lim
ln( 4)k
kx
k
132 2 lim
14
k
kx
k
4
2 2 lim 2 23k
kx x
k
2 325
2Check the endpoints!
0
2 2 ?
ln( 3)
kk
k
x
k
![Page 24: Power Series Radii and Intervals of Convergence. First some examples Consider the following example series: What does our intuition tell us about the.](https://reader035.fdocuments.net/reader035/viewer/2022070308/551c16fb550346a84f8b56bb/html5/thumbnails/24.jpg)
What about the convergence of
We start by setting up the ratio test limit.112 2
ln( 4)lim
2 2
ln( 3)
kk
kkk
x
k
x
k
112 2 ln( 3)lim
ln( 4)2 2
kk
kkk
x k
kx
ln( 3)2 2 lim
ln( 4)k
kx
k
132 2 lim
14
k
kx
k
4
2 2 lim 2 23k
kx x
k
2 325
2
c.c.Radius of Convergence is ½.Interval of convergence is [-5/2, -3/2)
0
2 2 ?
ln( 3)
kk
k
x
k