infinite sequences and series

11INFINITE SEQUENCES AND SERIESINFINITE SEQUENCES AND SERIES

In section 11.9, we were able to

find power series representations for

a certain restricted class of functions.

INFINITE SEQUENCES AND SERIES

Here, we investigate more general

problems.

Which functions have power series representations?

How can we find such representations?


11.10Taylor and Maclaurin Series


In this section, we will learn:

How to find the Taylor and Maclaurin Series of a function

and to multiply and divide a power series.

TAYLOR & MACLAURIN SERIES

We start by supposing that f is any function

that can be represented by a power series

20 1 2

3 43 4

( ) ( ) ( )

( ) ( ) ... | |

f x c c x a c x a

c x a c x a x a R

Equation 1


Let’s try to determine what the

coefficients cn must be in terms of f.

To begin, notice that, if we put x = a in Equation 1, then all terms after the first one are 0 and we get:

f(a) = c0


By Theorem 2 in Section 11.9, we can

differentiate the series in Equation 1 term by

term:

21 2 3

34

'( ) 2 ( ) 3 ( )

4 ( ) ... | |

f x c c x a c x a

c x a x a R

Equation 2


Substitution of x = a in Equation 2

gives:

f’(a) = c1


Now, we differentiate both sides of Equation 2

and obtain:

2 3

24

''( ) 2 2 3 ( )

3 4 ( ) ... | |

f x c c x a

c x a x a R

Equation 3


Again, we put x = a in Equation 3.

The result is:

f’’(a) = 2c2


Let’s apply the procedure

one more time.


Differentiation of the series in Equation 3

gives:

3 4

25

'''( ) 2 3 2 3 4 ( )

3 4 5 ( ) ... | |

f x c c x a

c x a x a R

Equation 4


Then, substitution of x = a in Equation 4

gives:

f’’’(a) = 2 · 3c3 = 3!c3


By now, you can see the pattern.

If we continue to differentiate and substitute x = a, we obtain:

( ) ( ) 2 3 4 !nn nf a nc n c


Solving the equation for the nth

coefficient cn, we get:

( ) ( )

!

n

n

f ac

n


The formula remains valid even for n = 0

if we adopt the conventions that 0! = 1

and f (0) = (f).

Thus, we have proved the following theorem.


If f has a power series representation

(expansion) at a, that is, if

then its coefficients are given by:

Theorem 5

0

( ) ( ) | |nn

n

f x c x a x a R

( ) ( )

!

n

n

f ac

n


Substituting this formula for cn back into

the series, we see that if f has a power series

expansion at a, then it must be of the following

form.

Equation 6

TAYLOR & MACLAURIN SERIES Equation 6( )

0

2

3

( )( ) ( )

!

'( ) ''( )( ) ( ) ( )

1! 2!'''( )

( )3!

nn

n

f af x x a

n

f a f af a x a x a

f ax a

TAYLOR SERIES

The series in Equation 6 is called

the Taylor series of the function f at a

(or about a or centered at a).

TAYLOR SERIES

For the special case a = 0, the Taylor series

becomes:

( )

0

2

(0)( )

!

'(0) ''(0)(0)

1! 2!

nn

n

ff x x

n

f ff x x

Equation 7

MACLAURIN SERIES

This case arises frequently

enough that it is given the special

name Maclaurin series.

Equation 7


The Taylor series is named after the English

mathematician Brook Taylor (1685–1731).

The Maclaurin series is named for the Scottish

mathematician Colin Maclaurin (1698–1746).

This is despite the fact that the Maclaurin series is really just a special case of the Taylor series.

MACLAURIN SERIES

Maclaurin series are named after Colin

Maclaurin because he popularized them

in his calculus textbook Treatise of Fluxions

published in 1742.


We have shown that if, f can be represented

as a power series about a, then f is equal to

the sum of its Taylor series.

However, there exist functions that are not equal to the sum of their Taylor series.

An example is given in Exercise 70.

Note


Find the Maclaurin series

of the function f(x) = ex and

its radius of convergence.

Example 1


If f(x) = ex, then f (n)(x) = ex.

So, f (n)(0) = e0 = 1 for all n.

Hence, the Taylor series for f at 0 (that is, the Maclaurin series) is:

Example 1

( ) 2 3

0 0

(0)1

! ! 1! 2! 3!

n nn

n n

f x x x xx

n n


To find the radius of convergence,

we let an = xn/n!

Then,

So, by the Ratio Test, the series converges for all x and the radius of convergence is R = ∞.

11 ! | |

0 1( 1)! 1

nn

nn

a x n x

a n x n


The conclusion we can draw from

Theorem 5 and Example 1 is:

If ex has a power series expansion at 0, then

0 !

nx

n

xe

n


So, how can we determine

whether ex does have a power

series representation?


Let’s investigate the more general

question:

Under what circumstances is a function equal to the sum of its Taylor series?


In other words, if f has derivatives of all

orders, when is the following true?

( )

0

( )( ) ( )

!

nn

n

f af x x a

n


As with any convergent series,

this means that f(x) is the limit of

the sequence of partial sums.


In the case of the Taylor series, the partial

sums are:

( )

0

2

( )

( )( ) ( )

!

'( ) ''( )( ) ( ) ( )

1! 2!

( )( )

!

ini

ni

nn

f aT x x a

i

f a f af a x a x a

f ax a

n

nTH-DEGREE TAYLOR POLYNOMIAL OF f AT a

Notice that Tn is a polynomial

of degree n called the nth-degree Taylor

polynomial of f at a.


For instance, for the exponential function

f(x) = ex, the result of Example 1 shows that

the Taylor polynomials at 0 (or Maclaurin

polynomials) with n = 1, 2, and 3 are:

2

1 2

2 3

3

( ) 1 ( ) 12!

( ) 12! 3!

xT x x T x x

x xT x x


The graphs of the exponential function

and those three Taylor polynomials are

drawn here.


In general, f(x) is the sum of its Taylor

series if:

( ) lim ( )nn

f x T x

REMAINDER OF TAYLOR SERIES

If we let Rn(x) = f(x) – Tn(x)

so that f(x) = Tn(x) + Rn(x)

then Rn(x) is called the remainder of

the Taylor series.


If we can somehow show that ,

then it follows that:

Therefore, we have proved the following.

lim ( ) 0nnR x

lim ( ) lim[ ( ) ( )]

( ) lim ( )

( )

n nn n

nn

T x f x R x

f x R x

f x


If f(x) = Tn(x) + Rn(x), where Tn is

the nth-degree Taylor polynomial of f at a

and

for |x – a| < R, then f is equal to the sum of

its Taylor series on the interval |x – a| < R.

Theorem 8

lim ( ) 0nnR x


In trying to show that

for a specific function f, we usually use

the following fact.

lim ( ) 0nnR x

TAYLOR’S INEQUALITY

If |f (n+1)(x)| ≤ M for |x – a| ≤ d,

then the remainder Rn(x) of the Taylor series

satisfies the inequality

Theorem 9

1| ( ) | | | for | |( 1)!

nn

MR x x a x a d

n


To see why this is true for n = 1, we

assume that |f’’(x)| ≤ M.

In particular, we have f’’(x) ≤ M.

So, for a ≤ x ≤ a + d, we have:

''( )x x

a af t dt M dt


An antiderivative of f’’ is f’.

So, by Part 2 of the Fundamental Theorem of Calculus (FTC2), we have:

f’(x) – f’(a) ≤ M(x – a) or

f’(x) ≤ f’(a) + M(x – a)


Thus,

2

2

'( ) [ '( ) ( )

( )( ) ( ) '( )( )

2

( ) ( ) '( )( ) ( )2

x x

a af t dt f a M t a dt

x af x f a f a x a M

Mf x f a f a x a x a


However,

R1(x) = f(x) – T1(x) = f(x) – f(a) – f’(a)(x – a)

So,2

1( ) ( )2

MR x x a


A similar argument, using f’’(x) ≥ -M, shows that:

So,

21( ) ( )

2

MR x x a

21| ( ) | | |

2

MR x x a


We have assumed that x > a.

However, similar calculations show that this inequality is also true for x < a.


This proves Taylor’s Inequality for

the case where n = 1.

The result for any n is proved in a similar way by integrating n + 1 times.

See Exercise 69 for the case n = 2


In Section 11.11, we will explore the use of

Taylor’s Inequality in approximating functions.

Our immediate use of it is in conjunction with

Theorem 8.

Note


In applying Theorems 8 and 9,

it is often helpful to make use of

the following fact.


This is true because we know from Example 1 that the series ∑ xn/n! converges for all x, and so its nth term approaches 0.

lim 0 for every real number!

n

n

xx

n

Equation 10


Prove that ex is equal to the sum of

its Maclaurin series.

If f(x) = ex, then f (n+1)(x) = ex for all n.

If d is any positive number and |x| ≤ d, then |f (n+1)(x)| = ex ≤ ed.

Example 2


So, Taylor’s Inequality, with a = 0 and

M = ed, says that:

Notice that the same constant M = ed works for every value of n.

1| ( ) | | | for | |( 1)!

dn

n

eR x x x d

n

Example 2


By Theorem 8, ex is equal to the sum

of its Maclaurin series, that is,

0

for all!

nx

n

xe x

n

E. g. 2—Equation 11


In particular, if we put x = 1 in Equation 11,

we obtain the following expression for

the number e as a sum of an infinite series:

0

1 1 1 11

! 1! 2! 3!n

en

Equation 12


Find the Taylor series for f(x) = ex

at a = 2.

We have f (n)(2) = e2. So, putting a = 2 in the definition of a Taylor series

(Equation 6), we get:

Example 3

( ) 22 2

0 0

(2)( 2) ( 2)

! !

n

n n

f ex x

n n


Again it can be verified, as in Example 1,

that the radius of convergence is R = ∞.

As in Example 2, we can verify

that lim ( ) 0nnR x



Thus,

2

0

( 2) for all!

x n

n

ee x x

n



We have two power series expansions for ex,

the Maclaurin series in Equation 11 and the

Taylor series in Equation 13.

The first is better if we are interested in values of x near 0.

The second is better if x is near 2.


Find the Maclaurin series for sin x

and prove that it represents sin x

for all x.

Example 4


We arrange our computation in two columns:

(4) (4)

( ) sin (0) 0

'( ) cos '(0) 1

''( ) sin ''(0) 0

'''( ) cos '''(0) 1

( ) sin (0) 0

f x x f

f x x f

f x x f

f x x f

f x x f

Example 4


As the derivatives repeat in a cycle of four,

we can write the Maclaurin series as follows:

2 3

3 5 7

2 1

0

'(0) ''(0) '''(0)(0)

1! 2! 3!

3! 5! 7!

( 1)(2 1)!

nn

n

f f ff x x x

x x xx

x

n

Example 4


Since f (n+1)(x) is ±sin x or ±cos x,

we know that |f (n+1)(x)| ≤ 1 for all x.

Example 4


By Equation 10, the right side of that inequality

approaches 0 as n → ∞.

So, |Rn(x)| → 0 by the Squeeze Theorem.

It follows that Rn(x) → 0 as n → ∞.

So, sin x is equal to the sum of its Maclaurin series by Theorem 8.

Example 4


We state the result of Example 4 for future

reference.

3 5 7

2 1

0

sin3! 5! 7!

( 1) for all(2 1)!

nn

n

x x xx x

xx

n

Equation 15


The figure shows the graph of sin x

together with its Taylor (or Maclaurin)

polynomials

1

3

3

3 5

5

( )

( )3!

3! 5!

T x x

xT x x

x xT x


Notice that, as n increases, Tn(x) becomes

a better approximation to sin x.


Find the Maclaurin series for cos x.

We could proceed directly as in Example 4.

However, it’s easier to differentiate the Maclaurin series for sin x given by Equation 15, as follows.

Example 5

TAYLOR & MACLAURIN SERIES Example 5

3 5 7

2 4 6

4 62

cos (sin )

3! 5! 7!

3 5 71

3! 5! 7!

12! 4! 6!

dx xdx

d x x xx

dx

x x x

x x x


The Maclaurin series for sin x converges

for all x.

So, Theorem 2 in Section 11.9 tells us that the differentiated series for cos x also converges for all x.

Example 5


Thus,

2 4 6

2

0

cos 12! 4! 6!

( 1) for all(2 )!

nn

n

x x xx

xx

n



The Maclaurin series for ex, sin x, and cos x

that we found in Examples 2, 4, and 5 were

discovered by Newton.

These equations are remarkable because they say we know everything about each of these functions if we know all its derivatives at the single number 0.


Find the Maclaurin series for the function

f(x) = x cos x.

Instead of computing derivatives and substituting in Equation 7, it’s easier to multiply the series for cos x (Equation 16) by x:

Example 6

2 2 1

0 0

cos ( 1) ( 1)(2 )! (2 )!

n nn

n n

x xx x x

n n


Represent f(x) = sin x as

the sum of its Taylor series

centered at π/3.

Example 7


Arranging our work in columns, we have:

3( ) sin

3 2

1'( ) cos '

3 2

3''( ) sin ''

3 2

1'''( ) cos '''

3 2

f x x f

f x x f

f x x f

f x x f

Example 7


That pattern repeats

indefinitely.

Example 7


Thus, the Taylor series at π/3 is:

2

3

2 3

' ''3 3

3 1! 3 2! 3

'''3

3! 3

3 1 3 1

2 2 1! 3 2 2! 3 2 3! 3

f ff x x

fx

x x x

Example 7


The proof that this series represents

sin x for all x is very similar to that in

Example 4.

Just replace x by x – π/3 in Equation 14.

Example 7


We can write the series in sigma notation

if we separate the terms that contain :

2

0

2 1

0

( 1) 3sin

2(2 )! 3

( 1)

2(2 1)! 3

nn

n

nn

n

x xn

xn

Example 7

3


We have obtained two different series

representations for sin x, the Maclaurin

series in Example 4 and the Taylor series

in Example 7.

It is best to use the Maclaurin series for values of x near 0 and the Taylor series for x near π/3.


Notice that the third Taylor polynomial T3

in the figure is a good approximation to sin x

near π/3 but not as good near 0.


Compare it with the third Maclaurin polynomial

T3 in the earlier figure—where the opposite is

true.


The power series that we obtained by indirect

methods in Examples 5 and 6 and in Section

11.9 are indeed the Taylor or Maclaurin series

of the given functions.


That is because Theorem 5 asserts that,

no matter how a power series representation

f(x) = ∑ cn(x – a)n is obtained, it is always true

that cn = f (n)(a)/n!

In other words, the coefficients are uniquely determined.



for f(x) = (1 + x)k, where k

is any real number.

Example 8


Arranging our work in columns, we have:

1

2

3

( ) ( )

( ) (1 ) (0) 1

'( ) (1 ) '(0)

''( ) ( 1)(1 ) ''(0) ( 1)

'''( ) ( 1)( 2)(1 ) '''(0) ( 1)( 2)

( 1) ( 1)(1 ) (0) ( 1) ( 1)

k

k

k

k

n k n n

f x x f

f x k x f k

f x k k x f k k

f x k k k x f k k k

f k k k n x f k k k n

Example 8

BINOMIAL SERIES

Thus, the Maclaurin series of f(x) = (1 + x)k

is:

This series is called the binomial series.

( )

0 0

(0) ( 1) ( 1)

! !

nn n

n n

f k k k nx x

n n

Example 8


If its nth term is an, then

1

1( 1) ( 1)( ) !

( 1)! ( 1) ( 1)

1| |

| | | | | | as11 1

n

n

n

n

a

a

k k k n k n x n

n k k k n x

kk n n

x x x nn

n

Example 8


Therefore, by the Ratio Test,

the binomial series converges if |x| < 1

and diverges if |x| > 1.

Example 8

BINOMIAL COEFFICIENTS.

The traditional notation for the coefficients

in the binomial series is:

These numbers are called the binomial coefficients.

( 1)( 2) ( 1)

!

k k k k k n

n n


The following theorem states that (1 + x)k

is equal to the sum of its Maclaurin series.

It is possible to prove this by showing that the remainder term Rn(x) approaches 0.

That, however, turns out to be quite difficult.

The proof outlined in Exercise 71 is much easier.

THE BINOMIAL SERIES

If k is any real number and |x| < 1,

then

Theorem 17

0

2

3

(1 )

( 1)1

2!( 1)( 2)

3!

k n

n

kx x

n

k kkx x

k k kx


Though the binomial series always converges

when |x| < 1, the question of whether or not

it converges at the endpoints, ±1, depends on

the value of k.

It turns out that the series converges at 1 if -1 < k ≤ 0 and at both endpoints if k ≥ 0.


Notice that, if k is a positive integer and n > k,

then the expression for contains a factor

(k – k).

So, for n > k.

This means that the series terminates and reduces to the ordinary Binomial Theorem when k is a positive integer.

0kn

kn



for the function

and its radius of convergence.

Example 9

1( )

4f x

x


We write f(x) in a form where we can use

the binomial series:

1/ 2

1 1

44 1

4

1 11

2 42 1

4

x x

x

x

Example 9


Using the binomial series with k = –½

and with x replaced by –x/4, we have:

1/ 2

12

0

1

4

11

2 4

1

2 4

n

n

x

x

x

n

Example 9


2312 2

33 512 2 2

3 51 12 2 2 2

1 11

2 2 4 2! 4

3! 4

1

! 4

n

x x

x

n x

n

Example 9


We know from Theorem 17 that this series converges when |–x/4| < 1, that is, |x| < 4.

So, the radius of convergence is R = 4.

2 32 3

1 1 1 3 1 3 51

2 8 2!8 3!8

1 3 5 (2 1)

!8n

n

x x x

nx

n

Example 9


For future reference, we collect some

important Maclaurin series that we have

derived in this section and Section 11.9,

in the following table.

IMPORTANT MACLAURIN SERIES

2 3

0

2 3

0

11 1

1

1! 1! 2! 3!

n

n

nx

n

x x x x Rx

x x x xe R

n

Table 1


2 1 3 5 7

0

2 2 4 6

0

2 1 3 5 71

0

sin ( 1)(2 1)! 3! 5! 7!

cos ( 1) 1(2 )! 2! 4! 6!

tan ( 1) 12 1 3 5 7

nn

n

nn

n

nn

n

x x x xx x R

n

x x x xx R

n

x x x xx x R

n

Table 1


2

0

3

( 1)(1 ) 1

2!

( 1)( 2)1

3!

k n

n

k k kx x kx x

n

k k kx R

Table 1

USES OF TAYLOR SERIES

One reason Taylor series are important

is that they enable us to integrate

functions that we couldn’t previously

handle.


In fact, in the introduction to this chapter,

we mentioned that Newton often integrated

functions by first expressing them as power

series and then integrating the series term

by term.


The function f(x) = ex2 can’t be integrated

by techniques discussed so far.

Its antiderivative is not an elementary function (see Section 7.5).

In the following example, we use Newton’s idea to integrate this function.


a. Evaluate ∫ e-x2dx as an infinite series.

b. Evaluate correct to within

an error of 0.001

Example 10

1 2

0

xe dx


First, we find the Maclaurin series

for f(x) = e-x2

It is possible to use the direct method.

However, let’s find it simply by replacing x with –x2

in the series for ex given in Table 1.

Example 10 a


Thus, for all values of x,

Example 10 a

22

0

2

0

2 4 6

( )

!

( 1)!

1 ...1! 2! 3!

nx

n

nn

n

xe

n

x

n

x x x


Now, we integrate term by term:

This series converges for all x because

the original series for e-x2 converges for all x.

22 4 6 2

3 5 7

2 1

1 ( 1)1! 2! 3! !

3 1! 5 2! 7 3!

( 1)(2 1) !

nx n

nn

x x x xe dx dx

n

x x xC x

x

n n

Example 10 a


The FTC gives:

Example 10 b

2

13 5 7 91

00

1 1 1 13 10 42 216

1 1 1 13 10 42 216

3 1! 5 2! 7 3! 9 4!

1

1

0.7475

x x x x xx dx x


The Alternating Series Estimation

Theorem shows that the error involved

in this approximation is less than

1 10.001

11 5! 1320

Example 10 b


Another use of Taylor series is illustrated

in the next example.

The limit could be found with l’Hospital’s Rule.

Instead, we use a series.


Evaluate

Using the Maclaurin series for ex, we have the following result.

Example 11

20

1lim

x

x

e x

x


This is because power series are continuous functions.

2 3

2 20 0

2 3 4

20

2 3

0

1 11! 2! 3!1

lim lim

2! 3! 4!lim

1 1lim

2 3! 4! 5! 2

x

x x

x

x

x x xx

e x

x x

x x x

x

x x x

Example 11

MULTIPLICATION AND DIVISION OF POWER SERIES

If power series are added or subtracted,

they behave like polynomials.

Theorem 8 in Section 11.2 shows this.

In fact, as the following example shows, they can also be multiplied and divided like polynomials.

MULTIPLICATION AND DIVISION OF POWER SERIES

In the example, we find only the first

few terms.

The calculations for the later terms become tedious.

The initial terms are the most important ones.

MULTIPLICATION AND DIVISION

Find the first three nonzero terms

in the Maclaurin series for:

a. ex sin x

b. tan x

Example 12


Using the Maclaurin series for ex and sin x

in Table 1, we have:

Example 12 a

2 3 3

sin 11! 2! 3! 3!

x x x x xe x x


We multiply these expressions, collecting

like terms just as for polynomials:

Example 12 a

2 31 12 6

316

2 3 41 12 6

3 41 16 6

2 313

1 x x x

x x

x x x x

x x

x x x


Thus,

Example 12 a

2 313sinxe x x x x


Using the Maclaurin series in Table 1,

we have:

Example 12 b

3 5

2 4

sin 3! 5!tancos

12! 4!

x xxx

xx xx


We use a procedure like long division:

Example 12 b

3 51 23 15

2 4 3 51 1 1 12 24 6 120

3 51 12 24

3 51 13 30

3 51 13 6

516

1

x x x

x x x x x

x x x

x x

x x

x


Thus,

Example 12 b

3 51 23 15tan x x x x


Although we have not attempted to

justify the formal manipulations used in

Example 12, they are legitimate.

There is a theorem that states

the following:

Suppose both f(x) = Σcnxn and g(x) = Σbnxn converge for |x| < R and the series are multiplied as if they were polynomials.

Then, the resulting series also converges for |x| < R and represents f(x)g(x).


For division, we require b0 ≠ 0.

The resulting series converges for sufficiently small x.


infinite sequences and series

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