9.1 power series. 0 e 0 2 what you will learn all continuous functions can be represented as a...
TRANSCRIPT
![Page 1: 9.1 Power Series. 0 e 0 2 What You Will Learn All continuous functions can be represented as a polynomial Polynomials are easy to integrate and differentiate](https://reader034.vdocument.in/reader034/viewer/2022051621/5697bf921a28abf838c8f1d0/html5/thumbnails/1.jpg)
QuickTime™ and a decompressor
are needed to see this picture.
9.1 Power Series9.1 Power Series
![Page 2: 9.1 Power Series. 0 e 0 2 What You Will Learn All continuous functions can be represented as a polynomial Polynomials are easy to integrate and differentiate](https://reader034.vdocument.in/reader034/viewer/2022051621/5697bf921a28abf838c8f1d0/html5/thumbnails/2.jpg)
00
ee
00
22
![Page 3: 9.1 Power Series. 0 e 0 2 What You Will Learn All continuous functions can be represented as a polynomial Polynomials are easy to integrate and differentiate](https://reader034.vdocument.in/reader034/viewer/2022051621/5697bf921a28abf838c8f1d0/html5/thumbnails/3.jpg)
What You Will Learn
• All continuous functions can be represented as a polynomial
• Polynomials are easy to integrate and differentiate
• Calculators use polynomials to calculate trig functions, logarithmic functions etc.
• Downfall of polynomial equivalent functions is that they have an infinite number of terms.
![Page 4: 9.1 Power Series. 0 e 0 2 What You Will Learn All continuous functions can be represented as a polynomial Polynomials are easy to integrate and differentiate](https://reader034.vdocument.in/reader034/viewer/2022051621/5697bf921a28abf838c8f1d0/html5/thumbnails/4.jpg)
This is an example of an This is an example of an infinite seriesinfinite series..
1
1
Start with a square one Start with a square one unit by one unit:unit by one unit:
1
21
21
4
1
4+
1
81
8+
1
161
16+
1
32 1
64
1
32+
1
64+ 1+⋅⋅⋅=
This series This series convergesconverges (approaches a limiting value.) (approaches a limiting value.)
Many series do not converge:Many series do not converge:1 1 1 1 1
1 2 3 4 5+ + + + +⋅⋅⋅=∞
→
![Page 5: 9.1 Power Series. 0 e 0 2 What You Will Learn All continuous functions can be represented as a polynomial Polynomials are easy to integrate and differentiate](https://reader034.vdocument.in/reader034/viewer/2022051621/5697bf921a28abf838c8f1d0/html5/thumbnails/5.jpg)
In an infinite series:In an infinite series: 1 2 31
n kk
a a a a a∞
=
+ + +⋅⋅⋅+ +⋅⋅⋅=∑
aa11, a, a22,…,… are are termsterms of the series. of the series. aann is the is the nnthth term term..
Partial sums:Partial sums: 1 1S a=
2 1 2S a a= +
3 1 2 3S a a a= + +
1
n
n kk
S a=
=∑
nnthth partial sum partial sum
If If SSnn has a limit as , then the series converges, has a limit as , then the series converges,
otherwise it otherwise it divergesdiverges..
n → ∞
→
![Page 6: 9.1 Power Series. 0 e 0 2 What You Will Learn All continuous functions can be represented as a polynomial Polynomials are easy to integrate and differentiate](https://reader034.vdocument.in/reader034/viewer/2022051621/5697bf921a28abf838c8f1d0/html5/thumbnails/6.jpg)
Geometric Series:
In a In a geometric seriesgeometric series, each term is found by multiplying , each term is found by multiplying
the preceding term by the same number, the preceding term by the same number, rr..
2 3 1 1
1
n n
n
a ar ar ar ar ar∞
− −
=
+ + + +⋅⋅⋅+ +⋅⋅⋅=∑
This converges to if , and diverges if .This converges to if , and diverges if .1
a
r−1r < 1r ≥
1 1r− < < is the is the interval of convergenceinterval of convergence..
→
![Page 7: 9.1 Power Series. 0 e 0 2 What You Will Learn All continuous functions can be represented as a polynomial Polynomials are easy to integrate and differentiate](https://reader034.vdocument.in/reader034/viewer/2022051621/5697bf921a28abf838c8f1d0/html5/thumbnails/7.jpg)
Geometric Series
n
n
1 - rS = a
1 - r
⎛ ⎞⎜ ⎟⎝ ⎠
Partial Sum of a Geometric Series:
Sn = a + ar + ar2 + ar3 + … + arn-1
-[r Sn = ar + ar2 + ar3 + … + arn
Sn – r Sn = a + arn
Sn (1 – r) = a (1 - rn)
![Page 8: 9.1 Power Series. 0 e 0 2 What You Will Learn All continuous functions can be represented as a polynomial Polynomials are easy to integrate and differentiate](https://reader034.vdocument.in/reader034/viewer/2022051621/5697bf921a28abf838c8f1d0/html5/thumbnails/8.jpg)
Sum of Converging Series
nn
nn
1 - rlim S = a , if r < 1, then r goes to zero and
1 - r
a S =
1 - rif r > 1, the series diverges.
Note: The interval of convergence is r < 1
→∞
⎛ ⎞⎜ ⎟⎝ ⎠
![Page 9: 9.1 Power Series. 0 e 0 2 What You Will Learn All continuous functions can be represented as a polynomial Polynomials are easy to integrate and differentiate](https://reader034.vdocument.in/reader034/viewer/2022051621/5697bf921a28abf838c8f1d0/html5/thumbnails/9.jpg)
Power Series Using Calculator
( )
102
x = 1
2
To calculate a partial sum of a power series on the calculator,
x
you can find the expanded form by entering:
seq x , x, 1, 10
to get 1 4 9 16 2
∑
{ }
( )( )2
5 36 49 64 81 100
and the sum by entering:
sum seq x , x, 1, 10
to get 385
![Page 10: 9.1 Power Series. 0 e 0 2 What You Will Learn All continuous functions can be represented as a polynomial Polynomials are easy to integrate and differentiate](https://reader034.vdocument.in/reader034/viewer/2022051621/5697bf921a28abf838c8f1d0/html5/thumbnails/10.jpg)
Example 1:Example 1:
3 3 3 3
10 100 1000 10000+ + + +⋅⋅⋅
.3 .03 .003 .0003+ + + +⋅⋅⋅ .333...=
1
3=
310
11
10−
aa
rr
3109
10
=3
9=
1
3=
→
![Page 11: 9.1 Power Series. 0 e 0 2 What You Will Learn All continuous functions can be represented as a polynomial Polynomials are easy to integrate and differentiate](https://reader034.vdocument.in/reader034/viewer/2022051621/5697bf921a28abf838c8f1d0/html5/thumbnails/11.jpg)
1 1 11
2 4 8− + − +⋅⋅⋅
11
12
⎛ ⎞− −⎜ ⎟⎝ ⎠
11
12
=+
13
2
=2
3=
aa
rr
Example 2:Example 2:
→
![Page 12: 9.1 Power Series. 0 e 0 2 What You Will Learn All continuous functions can be represented as a polynomial Polynomials are easy to integrate and differentiate](https://reader034.vdocument.in/reader034/viewer/2022051621/5697bf921a28abf838c8f1d0/html5/thumbnails/12.jpg)
The partial sum of a geometric series is:The partial sum of a geometric series is:( )1
1
n
n
a rS
r
−=
−
If thenIf then1r <( )1
lim1
n
n
a r
r→∞
−
− 1
a
r=
−
0
If and we let , then:If and we let , then:1x < r x=
2 31 x x x+ + + +⋅⋅⋅ 1
1 x=
−The more terms we use, the better our approximation The more terms we use, the better our approximation (over the interval of convergence.)(over the interval of convergence.)
→
![Page 13: 9.1 Power Series. 0 e 0 2 What You Will Learn All continuous functions can be represented as a polynomial Polynomials are easy to integrate and differentiate](https://reader034.vdocument.in/reader034/viewer/2022051621/5697bf921a28abf838c8f1d0/html5/thumbnails/13.jpg)
Example of a Power Series
The function y = 1
1 - x can be written as the power series:
Sn = 1 + x + x2 + x3 + ...
On your calculator, enter y1 =
1
1 - x and
y2 = sum seq xn , n, 0, 20( )( )
Then test a value: Enter y1(.5)
Enter y2(.5)
What do you notice?
![Page 14: 9.1 Power Series. 0 e 0 2 What You Will Learn All continuous functions can be represented as a polynomial Polynomials are easy to integrate and differentiate](https://reader034.vdocument.in/reader034/viewer/2022051621/5697bf921a28abf838c8f1d0/html5/thumbnails/14.jpg)
A power series is in this form:
c x c c x c x c x c xnn
nn
n
= + + + +⋅⋅⋅+ +⋅⋅⋅=
∞
∑ 0 1 22
33
0
or
c x a c c x a c x a c x a c x ann
nn
n
( ) ( ) ( ) ( ) ( )− = + − + − + − +⋅⋅⋅+ − +⋅⋅⋅=
∞
∑ 0 1 22
33
0
The coefficients c0, c1, c2… are constants.
The center “a” is also a constant.
(The first series would be centered at the origin if you graphed it. The second series would be shifted left or right. “a” is the new center.)
→
![Page 15: 9.1 Power Series. 0 e 0 2 What You Will Learn All continuous functions can be represented as a polynomial Polynomials are easy to integrate and differentiate](https://reader034.vdocument.in/reader034/viewer/2022051621/5697bf921a28abf838c8f1d0/html5/thumbnails/15.jpg)
Once we have a series that we know, we can find a new 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 series by doing the same thing to the left and right hand sides of the equation.sides of the equation.
This is a geometric series where This is a geometric series where r=-xr=-x..
1
x
x+To find a series forTo find a series for multiply both sides by multiply both sides by xx..
2 311
1x x x
x= − + − +⋅⋅⋅
+
2 3 4
1
xx x x x
x= − + − ⋅⋅⋅
+
1
1 x+Example 3:Example 3:
→
![Page 16: 9.1 Power Series. 0 e 0 2 What You Will Learn All continuous functions can be represented as a polynomial Polynomials are easy to integrate and differentiate](https://reader034.vdocument.in/reader034/viewer/2022051621/5697bf921a28abf838c8f1d0/html5/thumbnails/16.jpg)
Example 4:Example 4:
Given:Given: 2 311
1x x x
x= + + + +⋅⋅⋅
−find:find:
( )2
1
1 x−
1
1
d
dx x−( ) 11
dx
dx−
= − ( ) 21 1x
−= − − ⋅−
( )2
1
1 x=
−
So:So:( )
( )2 32
11
1
dx x x
dxx= + + + + ⋅⋅⋅
−
2 31 2 3 4x x x= + + + +⋅⋅⋅
We differentiated term by term.We differentiated term by term.→
![Page 17: 9.1 Power Series. 0 e 0 2 What You Will Learn All continuous functions can be represented as a polynomial Polynomials are easy to integrate and differentiate](https://reader034.vdocument.in/reader034/viewer/2022051621/5697bf921a28abf838c8f1d0/html5/thumbnails/17.jpg)
Example 5:Example 5:
Given:Given: 2 311
1x x x
x= − + − +⋅⋅⋅
+find:find: ( )ln 1 x+
( )1ln 1
1dx x c
x= + +
+∫
2 311
1t t t
t= − + − +⋅⋅⋅
+
hmm?hmm?
→
![Page 18: 9.1 Power Series. 0 e 0 2 What You Will Learn All continuous functions can be represented as a polynomial Polynomials are easy to integrate and differentiate](https://reader034.vdocument.in/reader034/viewer/2022051621/5697bf921a28abf838c8f1d0/html5/thumbnails/18.jpg)
Example 5:Example 5: 2 311
1t t t
t= − + − +⋅⋅⋅
+
( )2 3
0 0
11
1
x xdt t t t dt
t= − + − + ⋅⋅⋅
+∫ ∫
( ) 2 3 4
00
1 1 1ln 1
2 3 4
xx
t t t t t+ = − + − + ⋅⋅⋅
( ) ( ) 2 3 41 1 1ln 1 ln 1 0
2 3 4x x x x x+ − + = − + − + ⋅⋅⋅
( ) 2 3 41 1 1ln 1
2 3 4x x x x x+ = − + − + ⋅⋅⋅ 1 1x− < <
→
![Page 19: 9.1 Power Series. 0 e 0 2 What You Will Learn All continuous functions can be represented as a polynomial Polynomials are easy to integrate and differentiate](https://reader034.vdocument.in/reader034/viewer/2022051621/5697bf921a28abf838c8f1d0/html5/thumbnails/19.jpg)
( ) 2 3 41 1 1ln 1
2 3 4x x x x x+ = − + − + ⋅⋅⋅ 1 1x− < <
The previous examples of infinite series approximated The previous examples of infinite series approximated simple functions such as or .simple functions such as or .1
3
1
1 x−
This series would allow us to calculate a transcendental This series would allow us to calculate a transcendental function to as much accuracy as we like using only function to as much accuracy as we like using only pencil and paper!pencil and paper!
![Page 20: 9.1 Power Series. 0 e 0 2 What You Will Learn All continuous functions can be represented as a polynomial Polynomials are easy to integrate and differentiate](https://reader034.vdocument.in/reader034/viewer/2022051621/5697bf921a28abf838c8f1d0/html5/thumbnails/20.jpg)
Convergent Series
Only two kinds of series converge:
1) Geometric whose | r | < 1
2) Telescoping series
Example of a telescoping series: the middle terms cancel out
( )n 1 n = 1
n
1 1 1 = - n n + 1 n n + 1
1 1 1 1 1 1 1 = 1 - + - + - + - +...
2 2 3 3 4 4 5
= sum of 1 - last term
= 1 + lim
∞ ∞
=
→
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
∑ ∑
1 -
n + 1
= 1 + 0 = 1
∞
⎛ ⎞⎜ ⎟⎝ ⎠
![Page 21: 9.1 Power Series. 0 e 0 2 What You Will Learn All continuous functions can be represented as a polynomial Polynomials are easy to integrate and differentiate](https://reader034.vdocument.in/reader034/viewer/2022051621/5697bf921a28abf838c8f1d0/html5/thumbnails/21.jpg)
Finding a series for tan-1 x
1. Find a power series that represents on (-1,1)
2. Use integration to find a power series that represents
tan-1 x.
3. Graph the first four partial sums. Do the graphs suggest convergence on the open interval (-1, 1)?
4. Do you think that the series for tan-1 x converges at x = 1?
1 - x2 + x4 - x6 + ...+ (-1)nx2n + ...
1
1+ x2∫ dx = x - x3
3 +
x5
5 -
x7
7 +...+ (-1)n x2n+1
2n+1 +...
Yes to
4
1
1 + x2( )
Yes
![Page 22: 9.1 Power Series. 0 e 0 2 What You Will Learn All continuous functions can be represented as a polynomial Polynomials are easy to integrate and differentiate](https://reader034.vdocument.in/reader034/viewer/2022051621/5697bf921a28abf838c8f1d0/html5/thumbnails/22.jpg)
ex
Guess the function
Define a function f by a power series as follows:
2 3 4 nx x x xf (x) = 1 + x + + + +...+ + ...
2! 3! 4! n!
Find f ‘(x).
What function is this?