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:

 
(1)

Let's try to determine what the coefficients  must be in terms of . To begin, notice that if we put  in Equation 1, then all terms after the first one are  and we get



By Theorem 8.6.2, we can differentiate the series in Equation 1 term by term:

(2)

and substitution of 
 in Equation 2 gives  Now we differentiate both sides of Equation 2 and obtain

(3)

Again we put 
 in Equation 3. The result is

 

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

(4)

and substitution of 
 in Equation 4 gives



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

 

Solving this equation for the 
th coefficient , we get

 

This formula remains valid even for 
 if we adopt the conventions that  and .

Thus we have proved the following theorem.

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

(5)

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

(6)

The series in Equation 6 is called the Taylor series of the function 
 at  (or about  or centered at ). For the special case  the Taylor series becomes

(7)

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

NOTE: We have shown that if 
 can be represented as a power series about , then  is equal to the sum of its Taylor series. But there exist functions that are not equal to the sum of their Taylor series. For example, one can show that the function defined by



 is not equal to its Maclaurin series.

EXAMPLE 1: Find the Maclaurin series of the function 
 and its radius of convergence.


The conclusion we can draw from (5) and Example 1 is that if 
 has a power series expansion at , then

 

So how can we determine whether 
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 is it true that

 

As with any convergent series, this means that 
 is the limit of the sequence of partial sums. In the case of the Taylor series, the partial sums are



Notice that 
 is a polynomial of degree  called the th-degree Taylor polynomial of  at . For instance, for the exponential function , the result of Example 1 shows that the Taylor polynomials at  (or Maclaurin polynomials) with , and 3 are



                                             

In general, 
 is the sum of its Taylor series if

 

If we let



then 
 is called the remainder of the Taylor series. If we can somehow show that , then it follows that



We have therefore proved the following.

THEOREM: If 
, where  is the th-degree Taylor polynomial of  at  and

(8)

for 
, then  is equal to the sum of its Taylor series on the interval .

In trying to show that 
 for a specific function , we usually use the expression in the next theorem.

THEOREM (TAYLOR'S FORMULA): If 
 has  derivatives in an interval  that contains the number , then for  in  there is a number  strictly between  and  such that the remainder term in the Taylor series can be expressed as



NOTE 1: For the special case 
, if we put  and  in Taylor's Formula, we get

(9)

which is the Mean Value Theorem. In fact, Theorem 9 can be proved by a method similar to the proof of the Mean Value Theorem.

NOTE 2: Notice that the remainder term

(10)

is very similar to the terms in the Taylor series except that 
 is evaluated at  instead of at . All we say about the number  is that it lies somewhere between  and . The expression for  in Equation 10 is known as Lagrange's form of the remainder term.

NOTE 3: In Section 8.8 we will explore the use of Taylor's Formula in approximating 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:

(11)

This is true because we know from Example 1 that the series 
 converges for all  and so its th term approaches 0.

EXAMPLE 2: Prove that  is equal to the sum of its Taylor series with  (Maclaurin series).


From Example 2 it follows that

(12)

In particular, if we put 
 in Equation 12, we obtain

(13)

EXAMPLE 3: Find the Taylor series for 
 at .

Solution: We have 
 and so, putting  in the definition of a Taylor series (6), we get

 

Again it can be verified, as in Example 1, that the radius of convergence is 
. As in Example 2 we can verify that , so

(14)

We have two power series expansions for 
, the Maclaurin series in Equation 12 and the Taylor series in Equation 14. The first is better if we are interested in values of  near  and the second is better if  is near 2.

EXAMPLE 4: Find the Maclaurin series for 
 and prove that it represents  for all .

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:

 

Using the remainder term (10) with 
, we have



where 
 and  lies between 0 and . But  is  or . In any case,  and so

(15)

By Equation 11 the right side of this inequality approaches 
 as , so  by the Squeeze Theorem. It follows that  as , so  is equal to the sum of its Maclaurin series by Theorem 8. Thus

                   (16)

EXAMPLE 5: Find the Maclaurin series for 
.

Solution 1: 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:

 

Solution 2: We differentiate the Maclaurin series for 
 given by Equation 16:



Since the Maclaurin series for 
converges for all , Theorem 8.6.2 tells us that the differentiated series for  also converges for all . Thus

(17)

EXAMPLE 6: Find the Maclaurin series for .


REMARK: The power series that we obtained by indirect methods in Examples 5 and 6 and in Section 8.6 are indeed the Taylor or Maclaurin series of the given functions because Theorem 5 asserts that, no matter how we obtain a power series representation



it is always true that 
. In other words, the coefficients are uniquely determined.

EXAMPLE 7: Find the Maclaurin series for 
, where  is any real number.

Solution: Arranging our work in columns, we have

 

Therefore, the Maclaurin series of 
 is

 

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

 

Thus by the Ratio Test the binomial series converges if 
 and diverges if .

The traditional notation for the coefficients in the binomial series is

 

and these numbers are called the binomial coefficients. The following theorem states that 
 is equal to the sum of its Maclaurin series. It is possible to prove this by showing that the remainder term  approaches , but that turns out to be quite difficult.

THEOREM (THE BINOMIAL SERIES): If 
 is any real number and , then

(18)

Although the binomial series always converges when 
, the question of whether or not it converges at the endpoints, , depends on the value of . It turns out that the series converges at 1 if  and at both endpoints if . Notice that if  is a positive integer and , then the expression for  contains a factor , so



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

EXAMPLE 8: Find the Maclaurin series for 
 and its radius of convergence.


We collect in the following table, for future reference, some important Maclaurin series that we have derived in this section and the preceding one.



One reason that Taylor series are important is that they enable us to integrate functions that we couldn't previously handle. In fact, the function
 can't be integrated by techniques discussed so far because its antiderivative is not an elementary function (see Section 6.4). In the following example we write  as the Maclaurin series to integrate this function.

EXAMPLE 9:

(a) Evaluate 
 as an infinite series.

(b) Evaluate 
correct to within an error of 0.001.


Another use of Taylor series is illustrated in the next example. The limit could be found with l'Hospital's Rule, but instead we use a series.

EXAMPLE 10: Evaluate .