Power Series: Radii and Intervals of Convergence
Transcript of Power Series: Radii and Intervals of Convergence
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Power Series
Radii and Intervals of Convergence
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First some examplesConsider 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?
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Power SeriesNow 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
xk
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How does it go? We start by setting up the appropriate limit.
1!lim1 !
k
kk
x kkx
1
1 !lim
!
k
kk
xk
xk
lim
1k
xk
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
xk
Why the absolute values?Why on the x’s and not elsewhere?
The series is an example of a power series.0 !
k
k
xk
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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 xk
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
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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.
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Checking for ConvergenceI should use the ratio test. It is the test of choice when testing for convergence of power series!
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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 xk x
( 2)
lim( 1)k
k xk
The ratio test says that the series converges provided that this limit is less than 1. That is, when |x|<1.
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What about the convergence of
We start by setting up the ratio test limit.
132( 1) 1 !
lim3
2 1 !
k
kk
xkxk
13 (2 1)!lim(2 3)!3
k
kk
x kkx
2 3 4
0
3 3 3 3 1 31 ?
3! 5! 7! 9! 2 1 !
k k
k
x x x x xk
3
lim2 2 2 3k
xk 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 kk k k k k
Since the limit is 0 (which is less than 1), the ratio test says that the series converges absolutely for all x.
0
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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 xk
Don’t forget those absolute values!
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Now you work out the convergence of
We start by setting up the ratio test limit.
132( 1) 1
lim3
2 1
k
kk
xkxk
13 (2 1)lim(2 3)3
k
kk
x kkx
2 3 4
0
3 3 3 3 1 31
3 5 7 9 2 1
k k
k
x x x x xk
What does this tell us?
(2 1)3 lim(2 3)k
kxk
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?)
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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?
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01 when | - | the series converges absolutely.x xk
01 when | - | the series diverges.x xk
01 when | - | we don't know.x xk
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 !
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Recapping
01 when | - | the series converges absolutely.x xk
01 when | - | the series diverges.x xk
01 when | - | we don't know. x xk
0x 01xk
01x k
Must test endpoints separately!
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ConclusionsTheorem: If we have a power series ,
• It may converge only at x=x0.
00
( )nn
n
a x x
0x•It may converge for all x.
•It may converge on a finite interval centered at x=x0.
Radius of convergence is 0
Radius of conv. is infinite.
0x 0x R0x R
Radius of conv. is R.
0x
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ConclusionsTheorem: If we have a power series ,
• It may converge only at x=x0.
00
( )nn
n
a x x
0x•It may converge for all x.
•It may converge on a finite interval centered at x=x0.
0x 0x R0x R
0xWhat about absolute vs. conditional convergence?
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ConclusionsTheorem: If we have a power series ,
• It may converge only at x=x0.
00
( )nn
n
a x x
0x•It may converge for all x.
•It may converge on a finite interval centered at x=x0.
0x 0x R0x RWhat about absolute vs.
conditional convergence?