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MATH 1231MATHEMATICS 1B CALCULUS.
Section 5: - Power Series and Taylor Series.
The objective of this section is to become fa-miliar with the theory and application of powerseries and Taylor series.
By the end of this section students will be fa-miliar with:
convergence and divergence of power andTaylor series;
their importance;
their uses and applications.
In particular, students will be able to solve arange of problems that involve power and Tay-lor series.
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In general, a power series is a function of theform
k=0
a k( x a ) k
= a0 + a 1 ( x a ) + a 2 ( x a ) 2+ a3 ( x a ) 3 + a 4 ( x a ) 4 + . . . .where x is the variable; the ak are constants;and a is some xed number.
For example, consider the function
f ( x) = k=0
x3
k=
k=0
13
kxk .
This is a power series with a = 0, ak = (1 / 3)k
.What is the domain of this function? Thisquestion is equivalent to asking:
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We will see that power series are a really usefulway of representing some of the most impor-tant functions in mathematics, physics, chem-
istry and engineering.
One nice thing about power series is that theyare easy to differentiate and integrate. Forexample, consider
f ( x) = ex2 .The above function is not easy to integrateas its integral is not an elementary function.However, if we could write ex2 as a powerseries then we could just integrate each termof the series to get our answer.
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The domain of the function f represented bythe power series is called the
interval of convergence .
Half the length of this interval is called the
radius of convergence .
For a given power series k=0 a k( x a ) k thereare only three possibilities:
the series converges only when x = a
the series converges for all x
the series converges if |x a | < R and di-verges if |x a | > R , for some positive num-ber R .
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Ex. Study the behaviour of k=0
k!xk , and k=0
xk
k!.
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Ex. k=1
( x + 1) k log kk + 3
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Manipulation of Power Series:Given two power series in powers of ( x x0 ),f ( x) =
k=0a k( x x0 ) k , g( x) =
k=0bk( x x0 ) k
in which both converge for x in some intervalof convergence, then we can add, subtract ormultiply these two series and the resulting se-ries will also converge in that interval.
More importantly,
If f ( x) =
k=0
a k( x x0 ) k for |x x0 | < R then
i) f is continuous and differentiable for
|x x0 | < R andf ( x) =
k=0
ka k( x x0 ) k1 ;ii) f is integrable on |x x0 | < R and a
primitive for f is
F ( x) = k=0
a kk + 1
( x x0 ) k+1 .
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Ex. Consider the power series k=0
xk .
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Ex:Consider now, the series f ( x) = k=0
xk
k! which
we showed earlier converges for all x.
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From these examples we can nd the powerseries for other functions.
Ex. Find the power series for xex .
Ex. Write down the rst 4 terms of the powerseries for e
x1x and state where the series isvalid.
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Note that one can construct innite series (NOT
power series) in which the above results areNOT true.
Ex. Here is an example of a function denedin terms of a series which is continuous ev-erywhere but differentiable nowhere. f ( x) =
n =1
( 12
) n sin(4 n x).
As you have seen certain functions have rathernice power series. We stumbled on the powerseries for ex by considering a special example.How can we nd the power series for otherfunctions? For example, what about sin x?Does it have a simple power series and if sohow do we get it? We also would like to knowwhere such a series is valid? That is, for what
values of x does the series actually convergeback to the given function?
The answer to this question will be given byTaylors Theorem.
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Taylor Series:
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A power series for a given function in powersof x is called the Maclaurin Series.
We can now use this idea to obtain a formulafor the Maclaurin series for a given function.
Suppose the Maclaurin series is
f ( x) = a0 + a 1 x + a 2 x2 + a 3 x3 + ....
then
a 0 = f (0) , a 1 = f (0) , a 2 = f (0)
2! , a 3 =
f (0)3!
, . . .
This of course assumes that our function isdifferentiable innitely often.
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Ex. Find the Maclaurin series for sin x .
Ex. Find the Maclaurin series for cos x .
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Ex. Find the Maclaurin series for log(1 + x).
What happens if we try to nd the Maclau-rin series for log x in powers of x?
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Thus, we need a more general series for a func-tion called a Taylor series in which we try to ex-
press f ( x) as a polynomial in powers of ( xx0 ) .Denition: The power series
k=0
a k( x x0 ) k
is called the Taylor Series for f .
The co-efficients are given by
a 0 = f ( x0 ) , a 1 = f ( x0 ) , a 2 = f ( x0 )
2! , . . .
(Note that if x0 = 0, we just get a Maclau-rin Series for f .)
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Ex. Find the Taylor series for the functionlog( x) about the point x = 1.
We can see from the graphs (and it is truein general) that when we approximate a func-tion f by a nite number of terms of TaylorSeries in powers of x x0 , the closer the valueof x is to x0 the better the approximation. Wewill shortly be able to measure approximatelyhow close that will be.
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Ex: Find the Taylor series for the functioncos x about the point x = 2 , (i.e. in powers
of ( x 2 ).)
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Ex: An ancient formula for computing the squareroot (approximately) for a given number is
a 2 + x a + x2 a .This is simply the Maclaurin series for
f ( x) = a 2 + x.
Convergence of Taylor Series:
Two important questions arise. Firstly, exactlywhere does the Taylor series of a function con-verge? And secondly, if we truncate the Taylorseries for a function at some point, how goodan approximation is it?
The rst of these questions requires a knowl-edge of Complex Analysis and is beyond thescope of the course.
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To answer the second question we take thetruncated Taylor series (known as the Taylorpolynomial), with ( n + 1) terms:
P n ( x) =n
k=0
f ( k) ( x0 )( x x0 ) kk!
.
The remainder or error term, Rn ( x), is simplygiven by
R n ( x) = f ( x) P n ( x) .
How big can this remainder be? Is there someway of estimating it? The following result,
gives us a formula for the error involved in suchan approximation.
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Taylors Theorem: Suppose that f is (atleast) n + 1 times differentiable on the interval
( x0 a, x 0 + a ). Then given n , for each xI ,we can nd a c between x0 and x such that
R n ( x) = f ( n +1) ( c)( x x0 ) ( n +1)
( n + 1)! .
Corollary: If f is innitely differentiable and
for each point x with |x x0 | < R, we haveR n ( x) 0 as n , then f is represented byits Taylor series.
f ( x) = k=0
f ( k) ( x0 )( x x0 ) kk!
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Ex. Calculate the second and fourth degreeTaylor approximations to cos x about the pointx0 = 0 and estimate how accurate they are inthe range
1
2 x
1
2.
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Ex. Estimate the error involved when f ( x) = x is approximated by the rst 3 terms of itsTaylor series about x = 9 for x(8 , 10) .
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An Application to Max. and Min.
Ex: Consider the function f ( x) = cos( x6 ) . Thisclearly has a stationary point at x = 0 sincef ( x) = 6 x5 sin( x6 ) = 0 at x = 0.
Theorem: Suppose f is n times differentiableat x0 , and f ( x0 ) = 0.
If f ( n ) ( x0 ) = 0 for n = 2 , 3 ,...,k 1 and f ( k) ( x0 ) =0 then f has
a local minimum at x0 if k is even andf ( k) ( x0 ) > 0
a local maximum at x0 if k is even andf ( k) ( x0 ) < 0
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Indeterminate Forms:
A limit, which presents itself as 00 , is knownas an indeterminate form . We met these whenwe looked at LH opitals Rule. Using series,we can now see why LH opitals Rule actuallyworks and how to nd such limits using series.
Ex: limx0
sin xx
.
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Ex: limx1
log xx 1
.
limx0
2cos x
2 + x2
x4 .
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