11.8 Power Series 11.9 Representations of Functions as Power Series 11.10 Taylor and Maclaurin Series
11.8 Power Series
11.9Representations of Functions as
Power Series
11.10 Taylor and Maclaurin Series
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Power Series
A power series is a series of the form
where x is a variable and the cn’s are constants called the coefficients of the series.
A power series may converge for some values of x and diverge for other values of x.
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Power Series
The sum of the series is a function:
f (x) = c0 + c1x + c2x2 + . . . + cnxn + . . .
whose domain is the set of all x for which the series converges. Notice that f resembles a polynomial. The only difference is that f has infinitely many terms.
Note: if we take cn = 1 for all n, the power series becomes the geometric series
xn = 1 + x + x2 + . . . + xn + . . .
which converges when –1 < x < 1 and diverges when | x | 1.
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Power Series
More generally, a series of the form
is called a power series in (x – a) or a power series centered at a or a power series about a.
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Power Series
The main use of a power series is to provide a way to represent some of the most important functions that arise in mathematics, physics, and chemistry.
Example: the sum of the power series,
, is called a Bessel function.
•Electromagnetic waves in a cylindrical waveguide•Pressure amplitudes of inviscid rotational flows•Heat conduction in a cylindrical object•Modes of vibration of a thin circular (or annular) artificial membrane•Diffusion problems on a lattice•Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle•Solving for patterns of acoustical radiation•Frequency-dependent friction in circular pipelines•Signal processing
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Power Series
The first few partial sums are
Graph of the Bessel function:
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Power Series: convergence
The number R in case (iii) is called the radius of convergence of the power series.
This means: the radius of convergence is R = 0 in case (i) and R = in case (ii).
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Power Series
The interval of convergence of a power series is the interval of all values of x for which the series converges.
In case (i) the interval consists of just a single point a.
In case (ii) the interval is ( , ).
In case (iii) note that the inequality | x – a | < R can be rewritten as
a – R < x < a + R.
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Representations of Functions as Power Series
Example: Consider the series
We have obtained this equation by observing that the series is a geometric series with a = 1 and r = x.
We now regard Equation 1 as expressing the function
f (x) = 1/(1 – x) as a sum of a power series.
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Approximating Functions with Polynomials
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Example: Approximation of sin(x) near x = a
(1st order)(3rd order)
(5th order)
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Brook Taylor1685 - 1731
Brook Taylor was an accomplished musician and painter. He did research in a variety of areas, but is most famous for his development of ideas regarding infinite series.
Greg Kelly, Hanford High School, Richland, Washington
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Suppose we wanted to find a fourth degree polynomial of the form:
2 3 40 1 2 3 4P x a a x a x a x a x
ln 1f x x at 0x that approximates the behavior of
If we make , and the first, second, third and fourth derivatives the same, then we would have a pretty good approximation.
0 0P f
Practice:
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2 3 40 1 2 3 4P x a a x a x a x a x ln 1f x x
ln 1f x x
0 ln 1 0f
2 3 40 1 2 3 4P x a a x a x a x a x
00P a 0 0a
1
1f x
x
10 1
1f
2 31 2 3 42 3 4P x a a x a x a x
10P a 1 1a
2
1
1f x
x
10 1
1f
22 3 42 6 12P x a a x a x
20 2P a 2
1
2a
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2 3 40 1 2 3 4P x a a x a x a x a x ln 1f x x
3
12
1f x
x
0 2f
3 46 24P x a a x
30 6P a 3
2
6a
4
4
16
1f x
x
4 0 6f
4424P x a
440 24P a 4
6
24a
2
1
1f x
x
10 1
1f
22 3 42 6 12P x a a x a x
20 2P a 2
1
2a
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2 3 40 1 2 3 4P x a a x a x a x a x ln 1f x x
2 3 41 2 60 1
2 6 24P x x x x x
2 3 4
02 3 4
x x xP x x ln 1f x x
P x
f x
If we plot both functions, we see that near zero the functions match very well!
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This pattern occurs no matter what the original function was!
Our polynomial: 2 3 41 2 60 1
2 6 24x x x x
has the form: 42 3 40 0 0
0 02 6 24
f f ff f x x x x
or: 42 3 40 0 0 0 0
0! 1! 2! 3! 4!
f f f f fx x x x
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Maclaurin Series:
(generated by f at )0x
2 30 00 0
2! 3!
f fP x f f x x x
If we want to center the series (and it’s graph) at zero, we get the Maclaurin Series:
Taylor Series:
(generated by f at )x a
2 3
2! 3!
f a f aP x f a f a x a x a x a
Definition:
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2 3 4 5 61 0 1 0 1
1 0 2! 3! 4! 5! 6!
x x x x xP x x
cosy x
cosf x x 0 1f
sinf x x 0 0f
cosf x x 0 1f
sinf x x 0 0f
4 cosf x x 4 0 1f
2 4 6 8 10
1 2! 4! 6! 8! 10!
x x x x xP x
Exercise 1: find the Taylor polynomial approximation at 0 (Maclaurin series) for:
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cosy x 2 4 6 8 10
1 2! 4! 6! 8! 10!
x x x x xP x
The more terms we add, the better our approximation.
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To find Factorial using the TI-83:
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cos 2y xRather than start from scratch, we can use the function that we already know:
2 4 6 8 102 2 2 2 2
1 2! 4! 6! 8! 10!
x x x x xP x
Exercise 2: find the Taylor polynomial approximation at 0 (Maclaurin series) for:
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cos at 2
y x x
2 3
0 10 1
2 2! 2 3! 2P x x x x
cosf x x 02
f
sinf x x 12
f
cosf x x 02
f
sinf x x 12
f
4 cosf x x 4 02
f
3 5
2 2
2 3! 5!
x xP x x
Exercise 3: find the Taylor series for:
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When referring to Taylor polynomials, we can talk about number of terms, order or degree.
2 4
cos 12! 4!
x xx This is a polynomial in 3 terms.
It is a 4th order Taylor polynomial, because it was found using the 4th derivative.
It is also a 4th degree polynomial, because x is raised to the 4th power.
The 3rd order polynomial for is , but it is degree 2.cos x2
12!
x
The x3 term drops out when using the third derivative.
This is also the 2nd order polynomial.
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3) Use the fourth degree Taylor polynomial of cos(2x) to find the exact value of
Practice example:
.
1) Show that the Taylor series expansion of ex is:
2) Use the previous result to find the exact value of:
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Common Taylor Series:
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Properties of Power Series:Convergence
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3131
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Convergence of Power Series:
0
)(n
nn axc
1
lim
n
n
n c
cR
The center of the series is x = a. The series converges on the open interval and may converge at the endpoints. ),( RaRa
The Radius of Convergence for a power series is:
is
You must test each series that results at the endpoints of the interval separately for convergence.
Examples: The series is convergent on [-3,-1]
but the series is convergent on (-2,8].
02)1(
)2(
n
n
n
x
0 15
)3()1(
nn
nn
n
x
3333
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Convergence of Taylor Series: is
If f has a power series expansion centered at x = a, then the
power series is given by
And the series converges if and only if the Remainder satisfies:
0
)(
)(!
)()(
n
nn
axn
afxf
Where: is the remainder at x, (with c between x and a).
0)(lim
xRnn
1)!1(
)( )()()1(
n
ncf
n axxRn