TAYLOR AND MACLAURIN SERIES In the preceding section we were able to find power series representations for a certain

11.10 TAYLOR AND MACLAURIN SERIES In the preceding section we were able to find power series representations for a certain

restricted class of functions. Here we investigate more general problems: Which functions have power series representations? How can we find such representations?

We start by supposing that is any function that can be represented by a power series f

1 f 共x兲 苷 c 0 1 2 2 3 3 4 4 ⱍ x ⱍ

Let’s try to determine what the coefficients c n must be in terms of . To begin, notice that f if we put x苷a in Equation 1, then all terms after the first one are 0 and we get

f 共a兲 苷 c 0

By Theorem 11.9.2, we can differentiate the series in Equation 1 term by term:

2 f 共x兲 苷 c 1 2c 2 3 2 4c 4 3 ⱍ x ⱍ

and substitution of x苷a in Equation 2 gives

f 共a兲 苷 c 1

SECTION 11.10 TAYLOR AND MACLAURIN SERIES

Now we differentiate both sides of Equation 2 and obtain

3 f 共x兲 苷 2c 2 3 4 2 ⴢ 3c 2

Again we put x苷a in Equation 3. The result is

f 共a兲 苷 2c 2

Let’s apply the procedure one more time. Differentiation of the series in Equation 3 gives

4 f 共x兲 苷 2 ⴢ 3c 3 2 ⴢ 3 ⴢ 4c 4 5 2 ⱍ x ⱍ

and substitution of x苷a in Equation 4 gives

f 共a兲 苷 2 ⴢ 3c 3 苷 3!c 3

By now you can see the pattern. If we continue to differentiate and substitute x苷a , we obtain

f 共n兲 共a兲 苷 2 ⴢ 3 ⴢ 4 ⴢ ⴢ nc n 苷 n !c n

Solving this equation for the n th coefficient c n , we get

f 苷 共n兲 共a兲

This formula remains valid even for n苷 0 if we adopt the conventions that 0! 苷 1 and

f 共0兲 苷 f . Thus we have proved the following theorem.

5 THEOREM If has a power series representation (expansion) at , that is, if f a

f 共x兲 苷 兺 c n

n苷 0 ⱍ ⱍ

then its coefficients are given by the formula

f 共n兲

共a兲 n !

Substituting this formula for c n back into the series, we see that if f has a power series expansion at , then it must be of the following form. a

f 共n兲

6 f 共x兲 苷 兺 共a兲 n

n苷 0 n !

2 苷f f 共a兲 3 1!

共a兲

共a兲

CHAPTER 11 INFINITE SEQUENCES AND SERIES

TAYLOR AND MACLAURIN

The series in Equation 6 is called the Taylor series of the function f at a (or about a

The Taylor series is named after the English

or centered at a). For the special case a苷 0 the Taylor series becomes

mathematician Brook Taylor (1685–1731) and the Maclaurin series is named in honor of the Scot- tish mathematician Colin Maclaurin (1698–1746)

f 共x兲 苷 兺 x n 苷 f x 2

despite the fact that the Maclaurin series is

really just a special case of the Taylor series. But

the idea of representing particular functions as sums of power series goes back to Newton, and the general Taylor series was known to the Scot-

This case arises frequently enough that it is given the special name Maclaurin series.

tish mathematician James Gregory in 1668 and to the Swiss mathematician John Bernoulli in

NOTE We have shown that if f can be represented as a power series about , then is a f

the 1690s. Taylor was apparently unaware of the work of Gregory and Bernoulli when he published

equal to the sum of its Taylor series. But there exist functions that are not equal to the sum

his discoveries on series in 1715 in his book

of their Taylor series. An example of such a function is given in Exercise 70.

Methodus incrementorum directa et inversa. Maclaurin series are named after Colin Maclau-

V EXAMPLE 1 Find the Maclaurin series of the function f 共x兲 苷 e x and its radius of

rin because he popularized them in his calculus

convergence.

textbook Treatise of Fluxions published in 1742.

SOLUTION If f x , then f 共n兲

x , so f 共x兲 苷 e 共n兲 共x兲 苷 e 共0兲 苷 e 0 苷 1 for all . Therefore the n

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

f 共n兲

共0兲 n x

n苷 兺 0 0 n !

To find the radius of convergence we let a n 苷 x n 兾n! . Then

so, by the Ratio Test, the series converges for all and the radius of convergence x is . R苷

The conclusion we can draw from Theorem 5 and Example 1 is that if e x has a power

series expansion at 0, then

x n苷 兺 0 n ! So how can we determine whether e x 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 has derivatives of all orders, when f is it true that

f 共n兲 共a兲

f 共x兲 苷 n

n苷 兺 0 n !

As with any convergent series, this means that f 共x兲 is the limit of the sequence of partial

sums. In the case of the Taylor series, the partial sums are

f 共i兲 共a兲

共x兲 苷 i

i苷 兺 0 i !

f 共a兲 共n兲 共a兲 共a兲

苷f

SECTION 11.10 TAYLOR AND MACLAURIN SERIES

Notice that T n is a polynomial of degree called the nth-degree Taylor polynomial of f n

y=´

at a . For instance, for the exponential function f 共x兲 苷 e x , the result of Example 1 shows

y=T£(x)

that the Taylor polynomials at 0 (or Maclaurin polynomials) with n苷 1 , 2, and 3 are

y=T™(x)

y=T™(x)

(0, 1) y=T¡(x)

The graphs of the exponential function and these three Taylor polynomials are drawn in

Figure 1.

In general, f 共x兲 is the sum of its Taylor series if

y=T£(x)

N As increases, n T n 共x兲 appears to approach e x

in Figure 1. This suggests that e x is equal to the

n 共x兲

so that

f 共x兲 苷 T n

n 共x兲

sum of its Taylor series.

then R n 共x兲 is called the remainder of the Taylor series. If we can somehow show that lim nl R n 共x兲 苷 0 , then it follows that

We have therefore proved the following.

8 THEOREM If , where f 共x兲 苷 T n n 共x兲 T n is the n th-degree Taylor polyno-

mial of at and f a

for ⱍ x

ⱍ , then is equal to the sum of its Taylor series on the interval f

In trying to show that lim nl R n 共x兲 苷 0 for a specific function , we usually use the f following fact.

ⱍ , then the remainder f 共x兲 ⱍ ⱍ x ⱍ

9 TAYLOR’S INEQUALITY If for

R n 共x兲 of the Taylor series satisfies the inequality

n 共x兲 1 ⱍ for ⱍ x ⱍ ⱍ x ⱍ

To see why this is true for n 苷 1, we assume that

ⱍ . In particular, we have f 共x兲 ⱍ

f , so for a we have

dt

An antiderivative of f is , so by Part 2 of the Fundamental Theorem of Calculus, we f

have

f or

CHAPTER 11 INFINITE SEQUENCES AND SERIES

N As alternatives to Taylor’s Inequality, we have

Thus

the following formulas for the remainder term. If

f is continuous on an interval and I x僆I ,

then

共x兲 苷 n f

n ! y a 共t兲 dt

This is called the 2 integral form of the remainder f

term. Another formula, called Lagrange’s form of the remainder term, states that there is a number

z between and such x a that

This version is an extension of the Mean Value

Theorem (which is the case n苷 0 ).

Proofs of these formulas, together with dis-

A similar argument, using f , shows that

cussions of how to use them to solve the exam- ples of Sections 11.10 and 11.11, are given on the

website

www.stewartcalculus.com

Click on Additional Topics and then on Formulas

So

ⱍ 1 共x兲 ⱍ 2 ⱍ ⱍ

for the Remainder Term in Taylor series.

Although we have assumed that x

, similar calculations show that this inequality is

also true for x

This proves Taylor’s Inequality for the case where n苷 1 . The result for any n is proved

in a similar way by integrating n 1 times. (See Exercise 69 for the case n苷 2 .)

NOTE In Section 11.11 we will explore the use of Taylor’s Inequality in approxi- mating functions. Our immediate use of it is in conjunction with Theorem 8.

In applying Theorems 8 and 9 it is often helpful to make use of the following fact.

10 lim x

苷 0 for every real number x

nl

This is true because we know from Example 1 that the series

冘 n x 兾n! converges for all x

and so its n th term approaches 0.

V EXAMPLE 2 Prove that e x is equal to the sum of its Maclaurin series. SOLUTION If , then f x f 共x兲 苷 e x 共x兲 苷 e for all n . If d is any positive number and

, then . f 苷 e x 共x兲 d So Taylor’s Inequality, with a苷 0 and M苷e d ⱍ , ⱍ ⱍ ⱍ

says that

x n 1 for ⱍ x ⱍ ⱍ ⱍ ⱍ ⱍ

R n 共x兲

Notice that the same constant M苷e d works for every value of n. But, from Equa- tion 10, we have

lim x n 1 苷 e d lim nl ⱍ ⱍ nl 苷 0

SECTION 11.10 TAYLOR AND MACLAURIN SERIES

It follows from the Squeeze Theorem that lim nl ⱍ R n 共x兲 苷 0 ⱍ and therefore

lim

nl R n 共x兲 苷 0 for all values of x. By Theorem 8, e is equal to the sum of its

Maclaurin series, that is, x n

11 e x 苷 兺 for all x

n苷 0 n !

N In 1748 Leonard Euler used Equation 12 to

In particular, if we put x苷 1 in Equation 11, we obtain the following expression

find the value of correct to e 23 digits. In 2003

for the number as a sum of an infinite series: e

Shigeru Kondo, again using the series in (12), computed to more than 50 billion decimal e places. The special techniques employed to speed up the computation are explained on the

12 web page 苷 e苷

n苷 兺 0 n !

numbers.computation.free.fr

EXAMPLE 3 Find the Taylor series for

f x 共x兲 苷 e at a苷 2 .

SOLUTION We have f 共n兲

共2兲 苷 e 2 and so, putting a苷 2 in the definition of a Taylor series

(6), we get

f 共n兲

共2兲 n 苷

n苷 兺 0 n !

n苷 0 n !

Again it can be verified, as in Example 1, that the radius of convergence is R苷 . As in Example 2 we can verify that lim nl R n 共x兲 苷 0 , so

13 e x 苷

n苷 兺 0 for all x n !

We have two power series expansions for , the Maclaurin series in Equation 11 and x e the Taylor series in Equation 13. The first is better if we are interested in values of near x

0 and the second is better if is near 2. x EXAMPLE 4 Find the Maclaurin series for sin x and prove that it represents sin x for all . x

SOLUTION We arrange our computation in two columns as follows:

Since the derivatives repeat in a cycle of four, we can write the Maclaurin series as follows:

n苷 兺 0

CHAPTER 11 INFINITE SEQUENCES AND SERIES

N Figure 2 shows the graph of sin x together

Since f 共x兲 is sin x or cos x , we know that ⱍ f 共x兲 ⱍ 1 for all x. So we can

with its Taylor (or Maclaurin) polynomials

take in M苷 1 Taylor’s Inequality:

1 T 共x兲 苷 x x 3

n 1 苷 ⱍ x n T 1 3 14

共x兲 ⱍ

By Equation 10 the right side of this inequality approaches 0 as nl , so

Notice that, as increases, n T n 共x兲 becomes a

ⱍ R n 共x兲 l ⱍ 0 by the Squeeze Theorem. It follows that R n 共x兲 l 0 as nl , so sin x

better approximation to sin x .

is equal to the sum of its Maclaurin series by Theorem 8.

y T¡

We state the result of Example 4 for future reference.

T∞

y=sin x

兺 n for all x

EXAMPLE 5 Find the Maclaurin series for cos x . SOLUTION We could proceed directly as in Example 4 but it’s easier to differentiate the

Maclaurin series for sin x given by Equation 15:

dx 冉 3! 5! 7! 冊

cos x 苷

共sin x兲 苷 x

N The Maclaurin series for , e x sin x , and cos x

Since the Maclaurin series for sin x converges for all , Theorem 2 in Section 11.9 tells x

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

us that the differentiated series for cos x also converges for all . Thus x

covered, using different methods, 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.

兺 n for all x

n苷 0 共2n兲!

EXAMPLE 6 Find the Maclaurin series for the function f 共x兲 苷 x cos x . SOLUTION Instead of computing derivatives and substituting in Equation 7, it’s easier to

multiply the series for cos x (Equation 16) by : x

x 2n 苷

n x x cos x 苷 x 兺 兺 M

n苷 0 共2n兲! EXAMPLE 7 Represent f 共x兲 苷 sin x as the sum of its Taylor series centered at 兾3 .

n苷 0 共2n兲!

SECTION 11.10 TAYLOR AND MACLAURIN SERIES

SOLUTION Arranging our work in columns, we have

冉 s3

共x兲 苷 sin x f f 苷

共x兲 苷 cos x

N We have obtained two different series repre-

苷 s3

sentations 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

苷 x 1 near 0 and the Taylor series for near x 兾3 . f f

Notice that the third Taylor polynomial T 3 in Fig-

ure 3 is a good approximation to sin x near 兾3 but not as good near 0. Compare it with the third

and this pattern repeats indefinitely. Therefore the Taylor series at 兾3 is

Maclaurin polynomial T 3 in Figure 2, where the opposite is true.

冉 冊 x 冉 3 冊 1! 冉 3 冊 2! 冉 3 冊 3! 冉 3 冊

y=sin x

苷 s3 1 s3 2 1 3 x x x

The proof that this series represents sin x for all is very similar to that in Example 4. x

3 [Just replace by x x

兾3 in (14).] We can write the series in sigma notation if we

separate the terms that contain s3 :

T£

s3 n 2n

n苷 0 2 共2n兲! 冉 3 冊 n苷 0 2 冉 3 冊

sin x 苷 兺 x

FIGURE 3

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 because

Theorem 5 asserts that, no matter how a power series representation f 共x兲 苷 n 冘 c n

is obtained, it is always true that c n 苷 f 共n兲 共a兲兾n! . In other words, the coefficients are

uniquely determined.

EXAMPLE 8 Find the Maclaurin series for f k , where is any real number. k

SOLUTION Arranging our work in columns, we have

Therefore the Maclaurin series of f k is

f 共n兲 共0兲

n苷 兺 n苷 x 兺 0 0 n !

CHAPTER 11 INFINITE SEQUENCES AND SERIES

This series is called the binomial series. If its th term is n a n , then

Thus, by the Ratio Test, the binomial series converges if ⱍ x ⱍ and diverges if . ⱍ x ⱍ 1

The traditional notation for the coefficients in the binomial series is

and these numbers are called the binomial coefficients.

k is equal to the sum of its Maclaurin series. It is possible to prove this by showing that the remainder term R n 共x兲 approaches 0, but that

The following theorem states that

turns out to be quite difficult. The proof outlined in Exercise 71 is much easier.