Calculus I

Choi Nari

week 12 1
8. Series
8.6 Representing Functions as Power Series
❖ Representations of Functions using Geometric Series
➢ In this section we learn how to represent some familiar functions as sums of power series.

In Ex. 5 in Section 8.2, we obtained it by observing that the series is a geometric series with 𝑎 = 1 and 𝑟 = 𝑥.

➢ Now, we regard equation as expressing the function 𝑓 𝑥 = 1/(1 − 𝑥) as a sum of a power series. We say
that σ∞ 𝑛
𝑛=0 𝑥 , 𝑥 < 1 is a power series representation of 𝑓 𝑥 = 1/(1 − 𝑥) on the interval (-1,1).

Since the sum of a series is the limit of the sequence of partial sums, we have
1
= lim 𝑠𝑛 (𝑥) for 𝑆𝑛 𝑥 = 1 + 𝑥 + 𝑥 2 + ⋯ + 𝑥 𝑛
1−𝑥 𝑛→∞
Notices that as 𝑛 increases, 𝑆𝑛 (𝑥) becomes a better approximation
to 𝑓(𝑥) for −1 < 𝑥 < 1.

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8. Series
8.6 Representing Functions as Power Series
❖ Representations of Functions using Geometric Series
Example 1. Express 1/(1 + 𝑥 2 ) as the sum of a power series and find the interval of convergence.

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8. Series
8.6 Representing Functions as Power Series
❖ Representations of Functions using Geometric Series
Example 2. Find a power series representation of 1/(𝑥 + 2).

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8. Series
8.6 Representing Functions as Power Series
❖ Representations of Functions using Geometric Series
Example 3. Find a power series representation of 𝑥 3 /(𝑥 + 2).

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8. Series
8.6 Representing Functions as Power Series
❖ Differentiation and Integration of Power Series
➢ The sum of a power series is a function 𝑓 𝑥 = σ∞ 𝑛
𝑛=0(𝑥 − 𝑎) whose domain is the interval of convergence of
the series.
We want to differentiate and integrate such functions, and the following theorem says that we can do so
by differentiating or integrating each individual term in the series, just as we would for a polynomial.
This is called term-by-term differentiation and integration.

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8. Series
8.6 Representing Functions as Power Series
❖ Differentiation and Integration of Power Series

➢ Note 1: Equation (i) and (ii) in Thm 2 can be rewritten in the form

We know that, for finite sums, the derivative of a sum is the sum of the derivatives and the integral of a sum
is the sum of the integrals.
Above two equations assert that the same is true for infinite sums, provided we are dealing with power series.

➢ Note 2: Although Thm 2 says that the radius of convergence remains the same when a power series is differentiated
or integrated, this does not mean that the interval of convergence remains the same.

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8. Series
8.6 Representing Functions as Power Series
❖ Differentiation and Integration of Power Series
Example 5. Express 1/(1 − 𝑥)2 as a power series by differentiating eqn 1. What is the radius of convergence?

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8. Series
8.6 Representing Functions as Power Series
❖ Differentiation and Integration of Power Series
Example 6. Find a power series representation for ln(1 + 𝑥) and its radius of convergence.

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8. Series
8.6 Representing Functions as Power Series
❖ Differentiation and Integration of Power Series
Example 7. Find a power series representation for 𝑓 𝑥 = 𝑡𝑎𝑛−1 𝑥.

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8. Series
8.6 Representing Functions as Power Series
❖ Differentiation and Integration of Power Series
1
Example 8. Evaluate ‫׬‬ 𝑑𝑥 as a power series.
(1+𝑥 7 )

H.W.6,10,12,16,18,28,32,39,42
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8. Series
8.7 Taylor and Maclaurin Series
❖ Definitions of Taylor Series and Maclaurin Series

➢ Questions: Which functions have power series representations?

➢ Suppose that f is a function that can be represented by a power series

Let’s try to determine what the coefficients 𝑐𝑛 must be in terms of f. If we put 𝑥 = 𝑎 in eqn 1, then we get

By Thm 8.6.2, we can differentiate the series in eqn 1 term by term:

and substitution of 𝑥 = 𝑎 in eqn 2 gives

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8. Series
8.7 Taylor and Maclaurin Series
❖ Definitions of Taylor Series and Maclaurin Series

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

Solving this equation for the nth coefficient 𝑐𝑛 , we get

This formula remains valid even for n=0 if we adopt the conventions that 0!=1 and 𝑓 (0) = 𝑓.

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8. Series
8.7 Taylor and Maclaurin Series
❖ Definitions of Taylor Series and Maclaurin Series

➢ 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.
➢ The series in eqn 6 is called the
Taylor series of the function 𝑓 at a
(or about 𝑎 or centered at 𝑎).

8.7
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8. Series
8.7 Taylor and Maclaurin Series
❖ Definitions of Taylor Series and Maclaurin Series

➢ For the special case a=0 the Taylor series becomes

➢ Note 1: If 𝑓 can be represented as a power series about 𝑎, then 𝑓 is equal to the sum of its Taylor series.
But there exist functions that are not equal to the sum of their Taylor series.

➢ Note 2: The power series representation at 𝑎 of a function is unique, that is, it follows from Thm5 that if
𝑓 has a power series representation 𝑓 𝑥 = σ 𝑐𝑛 (𝑥 − 𝑎)𝑛 , then 𝑐𝑛 must be 𝑓 𝑛 (𝑎)/𝑛!.

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8. Series
8.7 Taylor and Maclaurin Series
❖ Definitions of Taylor Series and Maclaurin Series
Example 1. Find the Maclaurin series for the function 𝑓 𝑥 = 𝑒 𝑥 and its radius of convergence.
Soln.

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8. Series
8.7 Taylor and Maclaurin Series
❖ When is a Function Represented by Its Taylor Series?

➢ From Thm.5 and Ex.1, we can conclude that if we know that 𝑒 𝑥 has a power series representation at 0, then this
power series must be its Maclaurin series

So how can we determine whether 𝑒 𝑥 does have a power series representation?

➢ 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 it true that

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8. Series
8.7 Taylor and Maclaurin Series
❖ When Is a Function Represented by Its Taylor Series?

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

Note that 𝑇𝑛 is a polynomial of degree n called the nth-degree Taylor polynomial of 𝑓 at 𝑎.

➢ In general, 𝑓(𝑥) is the sum of its Taylor series if 𝑓 𝑥 = lim 𝑇𝑛 𝑥 .

➢ If lim 𝑅𝑛 𝑥 = 0, then lim 𝑇𝑛 𝑥 = lim 𝑓 𝑥 − 𝑅𝑛 𝑥 = 𝑓 𝑥 − lim 𝑅𝑛 𝑥 = 𝑓(𝑥).

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8. Series
8.7 Taylor and Maclaurin Series
❖ When Is a Function Represented by Its Taylor Series?

➢ In trying to show that lim 𝑅𝑛 𝑥 = 0 for a specific function 𝑓, we use the following Theorem.
𝑛→∞

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8. Series
8.7 Taylor and Maclaurin Series
❖ When Is a Function Represented by Its Taylor Series?

➢ When we apply Thm 8 and 9 it is helpful to make use of the following fact.

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8. Series
8.7 Taylor and Maclaurin Series
❖ When Is a Function Represented by Its Taylor Series?
Example 2. Prove that 𝑒 𝑥 is equal to the sum of its Taylor series.
Soln.

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8. Series
8.7 Taylor and Maclaurin Series
❖ When Is a Function Represented by Its Taylor Series?

➢ If we put x=1 in the equation, we obtain the following formula for the number e as a sum of an infinite series:

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8. Series
8.7 Taylor and Maclaurin Series
❖ When Is a Function Represented by Its Taylor Series?

Example 3. Find the Taylor series for 𝑓 𝑥 = 𝑒 𝑥 at 𝑎 = 2.

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8. Series
8.7 Taylor and Maclaurin Series
❖ Taylor Series of Important Functions
Example 4. Find the Maclaurin series for sin 𝑥 and prove it represents sin 𝑥 for all 𝑥.
Soln.

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8. Series
8.7 Taylor and Maclaurin Series
❖ Taylor Series of Important Functions
Example 4. Find the Maclaurin series for sin 𝑥 and prove it represents sin 𝑥 for all 𝑥.
Soln.

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8. Series
8.7 Taylor and Maclaurin Series
❖ Taylor Series of Important Functions
Example 5. Find the Maclaurin series for cos 𝑥.
Soln.

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8. Series
8.7 Taylor and Maclaurin Series
❖ Taylor Series of Important Functions
Example 6. Find the Maclaurin series for the function 𝑓 𝑥 = 𝑥 cos 𝑥.
Soln.

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8. Series
8.7 Taylor and Maclaurin Series
❖ Taylor Series of Important Functions
Example 7. Find the Maclaurin series for 𝑓 𝑥 = (1 + 𝑥)𝑘 , where k is any real number.
Soln.

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8. Series
8.7 Taylor and Maclaurin Series
❖ Taylor Series of Important Functions
Example 7. Find the Maclaurin series for 𝑓 𝑥 = (1 + 𝑥)𝑘 , where k is any real number.
Soln.

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8. Series
8.7 Taylor and Maclaurin Series
❖ Taylor Series of Important Functions

➢ The traditional notation for the coefficients in the binomial series is

and these numbers are called the binomial coefficients.

➢ The following theorem stated that (1 + 𝑥)𝑘 is equal to the sum of its Maclaurin series.

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8. Series
8.7 Taylor and Maclaurin Series
❖ Taylor Series of Important Functions
Example 8. Find the Maclaurin series for 𝑓 𝑥 = 1/ 4 − 𝑥 and its radius of convergence.
Soln.

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8. Series
8.7 Taylor and Maclaurin Series
❖ Taylor Series of Important Functions

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8. Series
8.7 Taylor and Maclaurin Series
❖ Taylor Series of Important Functions
Example 9. (a) Evaluate as an infinite series.
(b) Evaluate correct to within an error of 0.001.
Soln.

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8. Series
8.7 Taylor and Maclaurin Series
❖ Taylor Series of Important Functions
Example 9. (a) Evaluate as an infinite series.
(b) Evaluate correct to within an error of 0.001.
Soln.

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8. Series
8.7 Taylor and Maclaurin Series
❖ Taylor Series of Important Functions
Example 10. Evaluate lim (𝑒 𝑥 − 1 − 𝑥)/𝑥 2 .
𝑥→0
Soln.

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8. Series
8.7 Taylor and Maclaurin Series
❖ Multiplication and Division of Power Series
Example 11. Find the first three nonzero terms in the Maclaurin series for (a) 𝑒 𝑥 sin 𝑥 and (b) tan 𝑥.
Soln.

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8. Series
8.7 Taylor and Maclaurin Series
❖ Multiplication and Division of Power Series
Example 11. Find the first three nonzero terms in the Maclaurin series for (a) 𝑒 𝑥 sin 𝑥 and (b) tan 𝑥.
Soln.

H.W. 6,10,12,16,24,40,46,50,62
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