Learning Objectives
- 6.4.1Write the terms of the binomial series.
- 6.4.2Recognize the Taylor series expansions of common functions.
- 6.4.3Recognize and apply techniques to find the Taylor series for a function.
- 6.4.4Use Taylor series to solve differential equations.
- 6.4.5Use Taylor series to evaluate nonelementary integrals.
In the preceding section, we defined Taylor series and showed how to find the Taylor series for several common functions by explicitly calculating the coefficients of the Taylor polynomials. In this section we show how to use those Taylor series to derive Taylor series for other functions. We then present two common applications of power series. First, we show how power series can be used to solve differential equations. Second, we show how power series can be used to evaluate integrals when the antiderivative of the integrand cannot be expressed in terms of elementary functions. In one example, we consider an integral that arises frequently in probability theory.
The Binomial Series
Our first goal in this section is to determine the Maclaurin series for the function for all real numbers The Maclaurin series for this function is known as the binomial series. We begin by considering the simplest case: is a nonnegative integer. We recall that, for can be written as
The expressions on the right-hand side are known as binomial expansions and the coefficients are known as binomial coefficients. More generally, for any nonnegative integer the binomial coefficient of in the binomial expansion of is given by
(6.6)
and
(6.7)
For example, using this formula for we see that
We now consider the case when the exponent is any real number, not necessarily a nonnegative integer. If is not a nonnegative integer, then cannot be written as a finite polynomial. However, we can find a power series for Specifically, we look for the Maclaurin series for To do this, we find the derivatives of and evaluate them at
We conclude that the coefficients in the binomial series are given by
(6.8)
We note that if is a nonnegative integer, then the derivative is the zero function, and the series terminates. In addition, if is a nonnegative integer, then Equation 6.8 for the coefficients agrees with Equation 6.6 for the coefficients, and the formula for the binomial series agrees with Equation 6.7 for the finite binomial expansion. More generally, to denote the binomial coefficients for any real number we define
With this notation, we can write the binomial series for as
(6.9)
We now need to determine the interval of convergence for the binomial series Equation 6.9. We apply the ratio test. Consequently, we consider
Since
if and only if we conclude that the interval of convergence for the binomial series is The behavior at the endpoints depends on It can be shown that for the series converges at both endpoints; for the series converges at and diverges at and for the series diverges at both endpoints. The binomial series does converge to in for all real numbers but proving this fact by showing that the remainder is difficult.
Definition
For any real number the Maclaurin series for is the binomial series. It converges to for and we write
for
We can use this definition to find the binomial series for and use the series to approximate
Example 6.17
Finding Binomial Series
- Find the binomial series for
- Use the third-order Maclaurin polynomial to estimate Use Taylor’s theorem to bound the error. Use a graphing utility to compare the graphs of and
Solution
- Here Using the definition for the binomial series, we obtain
- From the result in part a. the third-order Maclaurin polynomial is
Therefore,
From Taylor’s theorem, the error satisfies
for some between and Since and the maximum value of on the interval occurs at we have
The function and the Maclaurin polynomial are graphed in Figure 6.10.Figure 6.10 The third-order Maclaurin polynomial provides a good approximation for for near zero.
Checkpoint 6.16
Find the binomial series for
Common Functions Expressed as Taylor Series
At this point, we have derived Maclaurin series for exponential, trigonometric, and logarithmic functions, as well as functions of the form In Table 6.1, we summarize the results of these series. We remark that the convergence of the Maclaurin series for at the endpoint and the Maclaurin series for at the endpoints and relies on a more advanced theorem than we present here. (Refer to Abel’s theorem for a discussion of this more technical point.)
Function | Maclaurin Series | Interval of Convergence |
---|---|---|
Table 6.1 Maclaurin Series for Common Functions
Earlier in the chapter, we showed how you could combine power series to create new power series. Here we use these properties, combined with the Maclaurin series in Table 6.1, to create Maclaurin series for other functions.
Example 6.18
Deriving Maclaurin Series from Known Series
Find the Maclaurin series of each of the following functions by using one of the series listed in Table 6.1.
Solution
- Using the Maclaurin series for we find that the Maclaurin series for is given by
This series converges to for all in the domain of that is, for all - To find the Maclaurin series for we use the fact that
Using the Maclaurin series for we see that the term in the Maclaurin series for is given by
For even, this term is zero. For odd, this term is Therefore, the Maclaurin series for has only odd-order terms and is given by
Checkpoint 6.17
Find the Maclaurin series for
We also showed previously in this chapter how power series can be differentiated term by term to create a new power series. In Example 6.19, we differentiate the binomial series for term by term to find the binomial series for Note that we could construct the binomial series for directly from the definition, but differentiating the binomial series for is an easier calculation.
Example 6.19
Differentiating a Series to Find a New Series
Use the binomial series for to find the binomial series for
Solution
The two functions are related by
so the binomial series for is given by
Checkpoint 6.18
Find the binomial series for
In this example, we differentiated a known Taylor series to construct a Taylor series for another function. The ability to differentiate power series term by term makes them a powerful tool for solving differential equations. We now show how this is accomplished.
Solving Differential Equations with Power Series
Consider the differential equation
Recall that this is a first-order separable equation and its solution is This equation is easily solved using techniques discussed earlier in the text. For most differential equations, however, we do not yet have analytical tools to solve them. Power series are an extremely useful tool for solving many types of differential equations. In this technique, we look for a solution of the form and determine what the coefficients would need to be. In the next example, we consider an initial-value problem involving to illustrate the technique.
Example 6.20
Power Series Solution of a Differential Equation
Use power series to solve the initial-value problem
Solution
Suppose that there exists a power series solution
Differentiating this series term by term, we obtain
If y satisfies the differential equation, then
Using Uniqueness of Power Series on the uniqueness of power series representations, we know that these series can only be equal if their coefficients are equal. Therefore,
Using the initial condition combined with the power series representation
we find that We are now ready to solve for the rest of the coefficients. Using the fact that we have
Therefore,
You might recognize
as the Taylor series for Therefore, the solution is
Checkpoint 6.19
Use power series to solve
We now consider an example involving a differential equation that we cannot solve using previously discussed methods. This differential equation
is known as Airy’s equation. It has many applications in mathematical physics, such as modeling the diffraction of light. Here we show how to solve it using power series.
Example 6.21
Power Series Solution of Airy’s Equation
Use power series to solve
with the initial conditions and
Solution
We look for a solution of the form
Differentiating this function term by term, we obtain
If y satisfies the equation then
Using Uniqueness of Power Series on the uniqueness of power series representations, we know that coefficients of the same degree must be equal. Therefore,
More generally, for we have In fact, all coefficients can be written in terms of and To see this, first note that Then
For we see that
Therefore, the series solution of the differential equation is given by
The initial condition implies Differentiating this series term by term and using the fact that we conclude that Therefore, the solution of this initial-value problem is
Checkpoint 6.20
Use power series to solve with the initial condition and
Evaluating Nonelementary Integrals
Solving differential equations is one common application of power series. We now turn to a second application. We show how power series can be used to evaluate integrals involving functions whose antiderivatives cannot be expressed using elementary functions.
One integral that arises often in applications in probability theory is Unfortunately, the antiderivative of the integrand is not an elementary function. By elementary function, we mean a function that can be written using a finite number of algebraic combinations or compositions of exponential, logarithmic, trigonometric, or power functions. We remark that the term “elementary function” is not synonymous with noncomplicated function. For example, the function is an elementary function, although not a particularly simple-looking function. Any integral of the form where the antiderivative of cannot be written as an elementary function is considered a nonelementary integral.
Nonelementary integrals cannot be evaluated using the basic integration techniques discussed earlier. One way to evaluate such integrals is by expressing the integrand as a power series and integrating term by term. We demonstrate this technique by considering
Example 6.22
Using Taylor Series to Evaluate a Definite Integral
- Express as an infinite series.
- Evaluate to within an error of
Solution
- The Maclaurin series for is given by
Therefore, - Using the result from part a. we have
The sum of the first four terms is approximately By the alternating series test, this estimate is accurate to within an error of less than
Checkpoint 6.21
Express as an infinite series. Evaluate to within an error of
As mentioned above, the integral arises often in probability theory. Specifically, it is used when studying data sets that are normally distributed, meaning the data values lie under a bell-shaped curve. For example, if a set of data values is normally distributed with mean and standard deviation then the probability that a randomly chosen value lies between and is given by
(6.10)
(See Figure 6.11.)
Figure 6.11 If data values are normally distributed with mean and standard deviation the probability that a randomly selected data value is between and is the area under the curve between and
To simplify this integral, we typically let This quantity is known as the score of a data value. With this simplification, integral Equation 6.10 becomes
(6.11)
In Example 6.23, we show how we can use this integral in calculating probabilities.
Example 6.23
Using Maclaurin Series to Approximate a Probability
Suppose a set of standardized test scores are normally distributed with mean and standard deviation Use Equation 6.11 and the first six terms in the Maclaurin series for to approximate the probability that a randomly selected test score is between and Use the alternating series test to determine how accurate your approximation is.
Solution
Since and we are trying to determine the area under the curve from to integral Equation 6.11 becomes
The Maclaurin series for is given by
Therefore,
Using the first five terms, we estimate that the probability is approximately By the alternating series test, we see that this estimate is accurate to within
Analysis
If you are familiar with probability theory, you may know that the probability that a data value is within two standard deviations of the mean is approximately Here we calculated the probability that a data value is between the mean and two standard deviations above the mean, so the estimate should be around The estimate, combined with the bound on the accuracy, falls within this range.
Checkpoint 6.22
Use the first five terms of the Maclaurin series for to estimate the probability that a randomly selected test score is between and Use the alternating series test to determine the accuracy of this estimate.
Another application in which a nonelementary integral arises involves the period of a pendulum. The integral is
An integral of this form is known as an elliptic integral of the first kind. Elliptic integrals originally arose when trying to calculate the arc length of an ellipse. We now show how to use power series to approximate this integral.
Example 6.24
Period of a Pendulum
The period of a pendulum is the time it takes for a pendulum to make one complete back-and-forth swing. For a pendulum with length that makes a maximum angle with the vertical, its period is given by
where is the acceleration due to gravity and (see Figure 6.12). (We note that this formula for the period arises from a non-linearized model of a pendulum. In some cases, for simplification, a linearized model is used and is approximated by Use the binomial series
to estimate the period of this pendulum. Specifically, approximate the period of the pendulum if
- you use only the first term in the binomial series, and
- you use the first two terms in the binomial series.
Figure 6.12 This pendulum has length and makes a maximum angle with the vertical.
Solution
We use the binomial series, replacing with Then we can write the period as
- Using just the first term in the integrand, the first-order estimate is
If is small, then is small. We claim that when is small, this is a good estimate. To justify this claim, consider
Since this integral is bounded by
Furthermore, it can be shown that each coefficient on the right-hand side is less than and, therefore, that this expression is bounded by
which is small for small. - For larger values of we can approximate by using more terms in the integrand. By using the first two terms in the integral, we arrive at the estimate
The applications of Taylor series in this section are intended to highlight their importance. In general, Taylor series are useful because they allow us to represent known functions using polynomials, thus providing us a tool for approximating function values and estimating complicated integrals. In addition, they allow us to define new functions as power series, thus providing us with a powerful tool for solving differential equations.
Section 6.4 Exercises
In the following exercises, use appropriate substitutions to write down the Maclaurin series for the given binomial.
174.
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In the following exercises, use the substitution in the binomial expansion to find the Taylor series of each function with the given center.
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at
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at
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at
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at (Hint:
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at
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at
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at
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at
In the following exercises, use the binomial theorem to estimate each number, computing enough terms to obtain an estimate accurate to an error of at most
186.
[T] using
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[T] using
In the following exercises, use the binomial approximation for to approximate each number. Compare this value to the value given by a scientific calculator.
188.
[T] using in
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[T] using in
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[T] using in
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[T] using in
192.
Integrate the binomial approximation of to find an approximation of
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[T] Recall that the graph of is an upper semicircle of radius Integrate the binomial approximation of up to order from to to estimate
In the following exercises, use the expansion to write the first five terms (not necessarily a quartic polynomial) of each expression.
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Use with to approximate
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Use the approximation for to approximate
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Find the derivative of at
201.
Find the th derivative at of
In the following exercises, find the Maclaurin series of each function.
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using the identity
209.
using the identity
In the following exercises, find the Maclaurin series of by integrating the Maclaurin series of term by term. If is not strictly defined at zero, you may substitute the value of the Maclaurin series at zero.
210.
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In the following exercises, compute at least the first three nonzero terms (not necessarily a quadratic polynomial) of the Maclaurin series of
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(see expansion for
In the following exercises, find the radius of convergence of the Maclaurin series of each function.
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230.
Find the Maclaurin series of
231.
Find the Maclaurin series of
232.
Differentiate term by term the Maclaurin series of and compare the result with the Maclaurin series of
233.
[T] Let and denote the respective Maclaurin polynomials of degree of and degree of Plot the errors for and compare them to on
234.
Use the identity to find the power series expansion of at (Hint: Integrate the Maclaurin series of term by term.)
235.
If find the power series expansions of and
236.
[T] Suppose that satisfies and Show that for all and that Plot the partial sum of on the interval
237.
[T] Suppose that a set of standardized test scores is normally distributed with mean and standard deviation Set up an integral that represents the probability that a test score will be between and and use the integral of the degree Maclaurin polynomial of to estimate this probability.
238.
[T] Suppose that a set of standardized test scores is normally distributed with mean and standard deviation Set up an integral that represents the probability that a test score will be between and and use the integral of the degree Maclaurin polynomial of to estimate this probability.
239.
[T] Suppose that converges to a function such that and Find a formula for and plot the partial sum for on
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[T] Suppose that converges to a function such that and Find a formula for and plot the partial sum for on
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Suppose that converges to a function such that where and Find a formula that relates and and compute
242.
Suppose that converges to a function such that where and Find a formula that relates and and compute
The error in approximating the integral by that of a Taylor approximation is at most In the following exercises, the Taylor remainder estimate guarantees that the integral of the Taylor polynomial of the given order approximates the integral of with an error less than
- Evaluate the integral of the appropriate Taylor polynomial and verify that it approximates the CAS value with an error less than
- Compare the accuracy of the polynomial integral estimate with the remainder estimate.
243.
[T] (You may assume that the absolute value of the ninth derivative of is bounded by
244.
[T] (You may assume that the absolute value of the derivative of is less than
The following exercises deal with Fresnel integrals.
245.
The Fresnel integrals are defined by and Compute the power series of and and plot the sums and of the first nonzero terms on
246.
[T] The Fresnel integrals are used in design applications for roadways and railways and other applications because of the curvature properties of the curve with coordinates Plot the curve for the coordinates of which were computed in the previous exercise.
247.
Estimate by approximating using the binomial approximation
248.
[T] Use Newton’s approximation of the binomial to approximate as follows. The circle centered at with radius has upper semicircle The sector of this circle bounded by the -axis between and and by the line joining corresponds to of the circle and has area This sector is the union of a right triangle with height and base and the region below the graph between and To find the area of this region you can write and integrate term by term. Use this approach with the binomial approximation from the previous exercise to estimate
249.
Use the approximation to approximate the period of a pendulum having length meters and maximum angle where Compare this with the small angle estimate
250.
Suppose that a pendulum is to have a period of seconds and a maximum angle of Use to approximate the desired length of the pendulum. What length is predicted by the small angle estimate
251.
Evaluate in the approximation to obtain an improved estimate for
252.
[T] An equivalent formula for the period of a pendulum with amplitude is where is the pendulum length and is the gravitational acceleration constant. When we get Integrate this approximation to estimate in terms of and Assuming meters per second squared, find an approximate length such that seconds.