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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

an inection point at x0 if k is odd.25


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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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