Calculus AB and Calculus BC

CHAPTER 10 Sequences and Series

C. POWER SERIES

C1. Definitions; Convergence.

An expression of the form

where the a’s are constants, is called a power series in x; and

is called a power series in (x − a).

If in (1) or (2) x is replaced by a specific real number, then the power series becomes a series of constants that either converges or diverges. Note that series (1) converges if x = 0 and series (2) converges if x = a.

RADIUS AND INTERVAL OF CONVERGENCE

If power series (1) converges when |x| < r and diverges when |x| > r, then r is called the radius of convergence. Similarly, r is the radius of convergence of power series (2) if (2) converges when |x − a| < r and diverges when |x − a| > r.

The set of all values of x for which a power series converges is called its interval of convergence. To find the interval of convergence, first determine the radius of convergence by applying the Ratio Test to the series of absolute values. Then check each endpoint to determine whether the series converges or diverges there.

EXAMPLE 33

Find all x for which the following series converges:

SOLUTION: By the Ratio Test, the series converges if

Thus, the radius of convergence is 1. The endpoints must be tested separately since the Ratio Test fails when the limit equals 1. When x = 1, (3) becomes 1 + 1 + 1 + · · · and diverges; when x = −1, (3) becomes 1−1 + 1−1 + ··· and diverges. Thus the interval of convergence is −1 < x < 1.

EXAMPLE 34

For what x does converge?

SOLUTION:

The radius of convergence is 1. When x = 1, we have an alternating convergent series; when x = −1, the series is which diverges. Thus, the series converges if − 1 < x 1.

EXAMPLE 35

For what values of x does converge?

SOLUTION:

which is always less than 1. Thus the series converges for all x.

EXAMPLE 36

Find all x for which the following series converges:

SOLUTION:

which is less than 1 if |x − 2| < 2, that is, if 0 < x < 4. Series (4) converges on this interval and diverges if |x − 2| > 2, that is, if x < 0 or x > 4.

When x = 0, (4) is 1 − 1 + 1 − 1 + · · · and diverges. When x = 4, (4) is 1 + 1 + 1 + · · · and diverges. Thus, the interval of convergence is 0 < x < 4.

EXAMPLE 37

Find all x for which the series converges.

SOLUTION: converges only at x = 0, since

unless x = 0.

C2. Functions Defined by Power Series.

Let the function f be defined by

its domain is the interval of convergence of the series.

Functions defined by power series behave very much like polynomials, as indicated by the following properties:

PROPERTY 2a. The function defined by (1) is continuous for each x in the interval of convergence of the series.

PROPERTY 2b. The series formed by differentiating the terms of series (1) converges to f ′(x) for each x within the radius of convergence of (1); that is,

Note that power series (1) and its derived series (2) have the same radius of convergence but not necessarily the same interval of convergence.

EXAMPLE 38

Let

Find the intervals of convergence of the power series for f (x) and f ′(x).

SOLUTION:

also,

and

Hence, the power series for f converges if −1 x 1.

For the derivative

also,

and

Hence, the power series for f ′ converges if −1 x < 1.

Thus, the series given for f (x) and f ′(x) have the same radius of convergence, but their intervals of convergence differ.

PROPERTY 2c. The series obtained by integrating the terms of the given series (1) converges to for each x within the interval of convergence of (1); that is,

EXAMPLE 39

Let Show that the power series for converges for all values of x in the interval of convergence of the power series for f (x).

SOLUTION: Obtain a series for by long division.

Then,

It can be shown that the interval of convergence is −1 < x < 1.

Then by Property 2c

Since when x = 0 we see that c = 1, we have

Note that this is a geometric series with ratio r = x and with a = 1; if |x| < 1, its sum is

C3. Finding a Power Series for a Function: Taylor and Maclaurin Series.

If a function f (x) is representable by a power series of the form

c₀ + c₁ (x − a) + c₂ (x − a)² + ··· + c_n (x − a)ⁿ +···

on an interval |x − a| < r, then the coefficients are given by

Taylor series

The series

is called the Taylor series of the function f about the number a. There is never more than one power series in (x − a) for f (x). It is required that the function and all its derivatives exist at x = a if the function f (x) is to generate a Taylor series expansion.

When a = 0 we have the special series

called the Maclaurin series of the function f; this is the expansion of f about x = 0.

EXAMPLE 40

Find the Maclaurin series for f (x) = e^x.

SOLUTION: Here f ′(x) = e^x, …, f ⁽ⁿ⁾ (x) = e^x, …, for all n. Then

f ′(0) = 1, …, f ⁽ⁿ⁾ (0) = 1, …,

for all n, making the coefficients

EXAMPLE 41

Find the Maclaurin expansion for f (x) = sin x.

SOLUTION:

Thus,

EXAMPLE 42

Find the Maclaurin series for

SOLUTION:

Then

Note that this agrees exactly with the power series in x obtained by different methods in Example 39.

EXAMPLE 43

Find the Taylor series for the function f (x) = ln x about x = 1.

SOLUTION:

Then

COMMON MACLAURIN SERIES

We list here for reference some frequently used Maclaurin series expansions together with their intervals of convergence:

FUNCTIONS THAT GENERATE NO SERIES.

Note that the following functions are among those that fail to generate a specific series in (x − a) because the function and/or one or more derivatives do not exist at x = a:

C4. Approximating Functions with Taylor and Maclaurin Polynomials.

The function f (x) at the point x = a is approximated by a Taylor polynomial P_n (x) of order n:

The Taylor polynomial P_n (x) and its first n derivatives all agree at a with f and its first n derivatives. The order of a Taylor polynomial is the order of the highest derivative, which is also the polynomial’s last term.

In the special case where a = 0, the Maclaurin polynomial of order n that approximates f (x) is

The Taylor polynomial P₁ (x) at x = 0 is the tangent-line approximation to f (x) near zero given by

f (x) P₁ (x) = f (0) + f ′(0)x.

It is the “best” linear approximation to f at 0, discussed at length in Chapter 4 §L.

A NOTE ON ORDER AND DEGREE

A Taylor polynomial has degree n if it has powers of (x − a) up through the nth. If f ⁽ⁿ⁾ (a) = 0, then the degree of P_n (x) is less than n. Note, for instance, in Example 45, that the second-order polynomial P₂ (x) for the function sin x (which is identical with P₁ (x)) is or just x, which has degree 1, not 2.

EXAMPLE 44

Find the Taylor polynomial of order 4 at 0 for f (x) = e^−x. Use this to approximate f (0.25).

SOLUTIONS: The first four derivatives are −e^−x, e^−x, −e^−x and e^−x ; at a = 0, these equal −1, 1,−1, and 1 respectively. The approximating Taylor polynomial of order 4 is therefore

With x = 0.25 we have

This approximation of e^−0.25 is correct to four places.

In Figure N10–2 we see the graphs of f (x) and of the Taylor polynomials:

FIGURE N10–2

Notice how closely P₄ (x) hugs f (x) even as x approaches 1. Since the series can be shown to converge for x > 0 by the Alternating Series Test, the error in P₄ (x) is less than the magnitude of the first omitted term, at x = 1. In fact, P₄ (1) = 0.375 to three decimal places, close to e⁻¹ ≈ 0.368.

EXAMPLE 45

(a) Find the Taylor polynomials P₁, P₃, P₅, and P₇ at x = 0 for f (x) = sin x.

(b) Graph f and all four polynomials in [−2π,2π] × [−2,2].

(c) Approximate sin using each of the four polynomials.

SOLUTIONS:

(a) The derivatives of the sine function at 0 are given by the following table:

From the table we know that

(b) Figure N10–3a shows the graphs of sin x and the four polynomials. In Figure N10–3b we see graphs only of sin x and P₇ (x), to exhibit how closely P₇ “follows” the sine curve.

FIGURE N10–3a

FIGURE N10–3b

(c) To four decimal places, sin = 0.8660. Evaluating the polynomials at we get

We see that P₇ is correct to four decimal places.

EXAMPLE 46

(a) Find the Taylor polynomials of degrees 0, 1, 2, and 3 generated by f (x) = ln x at x = 1.

(b) Graph f and the four polynomials on the same set of axes.

(c) Using P₂, approximate ln 1.3, and find a bound on the error.

SOLUTIONS:

(a) The derivatives of ln x at x = 1 are given in the table:

From the table we have

(b) Figure N10–4 shows the graphs of ln x and the four Taylor polynomials above, in [0,2.5] × [−1,1].

FIGURE N10–4

(c) ln 1.3 ≈ P₂ (1.3) = (1.3 − 1) − − 0.045 = 0.255.

For x = 1.3 the Taylor series converges by the Alternating Series Test, so the error is less than the magnitude of the first omitted term:

EXAMPLE 47

For what positive values of x is the approximate formula

ln (1 + x) =

correct to three decimal places?

SOLUTION: We can use series (4) of Common Maclaurin Series:

For x > 0, this is an alternating series with terms decreasing in magnitude and approaching 0, so the error committed by using the first two terms is less than If then the given approximation formula will yield accuracy to three decimal places. We therefore require that |x|³ < 0.0015 or that |x| < 0.114.

C5. Taylor’s Formula with Remainder; Lagrange Error Bound.

When we approximate a function using a Taylor polynomial, it is important to know how large the remainder (error) may be. If at the desired value of x the Taylor series is alternating, this issue is easily resolved: the first omitted term serves as an upper bound on the error. However, when the approximation involves a nonnegative Taylor series, placing an upper bound on the error is more difficult. This issue is resolved by the Lagrange remainder.

TAYLOR’S THEOREM. If a function f and its first (n + 1) derivatives are continuous on the interval |x − a| < r, then for each x in this interval

where

and c is some number between a and x. R_n (x) is called the Lagrange remainder.

Note that the equation above expresses f (x) as the sum of the Taylor polynomial P_n (x) and the error that results when that polynomial is used as an approximation for f (x).

When we truncate a series after the (n + 1)st term, we can compute the error bound R_n, according to Lagrange, if we know what to substitute for c. In practice we find, not R_n exactly, but only an upper bound for it by assigning to c the value between a and x that determines the largest possible value of R_n. Hence:

the Lagrange error bound.

EXAMPLE 48

Estimate the error in using the Maclaurin series generated by e^x to approximate the value of e.

SOLUTION: From Example 40 we know that f (x) = e^x generates the Maclaurin series

The Lagrange error bound is

To estimate e, we use x = 1. For 0 < c < 1, the maximum value of e^c is e. Thus:

EXAMPLE 49

Find the Maclaurin series for ln (1 + x) and the associated Lagrange error bound.

SOLUTION:

Then

where the Lagrange error bound is

NOTE: For 0 < x < 1 the Maclaurin series is alternating, and the error bound simplifies to the first omitted term. The more difficult Lagrange error bound applies for −1 < x <0.

EXAMPLE 50

Find the third-degree Maclaurin polynomial for and determine the upper bound on the error in estimating f (0.1).

SOLUTION: We first make a table of the derivatives, evaluated at x = 0 and giving us the coefficients.

Thus

Since this is not an alternating series for x = 0.1, we must use the Lagrange error bound:

where x = 0.1 and 0 < c < 0.1.

Note that is decreasing on the interval 0 < c < 0.1, so its maximum value occurs at c = 0. Hence:

C6. Computations with Power Series.

The power series expansions of functions may be treated as any other functions for values of x that lie within their intervals of convergence. They may be added, subtracted, multiplied, divided (with division by zero to be avoided), differentiated, or integrated. These properties provide a valuable approach for many otherwise difficult computations. Indeed, power series are often very useful for approximating values of functions, evaluating indeterminate forms of limits, and estimating definite integrals.

EXAMPLE 51

Compute to four decimal places.

SOLUTION: We can use the Maclaurin series,

and let to get

Note that, since this series converges by the Alternating Series Test, R₅ is less than the first term dropped:

so correct to four decimal places.

EXAMPLE 52

Estimate the error if the approximate formula

is used and |x| < 0.02.

SOLUTION: We obtain the first few terms of the Maclaurin series generated by

Then

Note that for x < 0, the series is not alternating, so we must use the Lagrange error bound. Here where −0.02 < c < 0.02. With |x| < 0.02, we see that the upper bound uses c = −0.02:

EXAMPLE 53

Use a series to evaluate

SOLUTION: From series (1) in Common Maclaurin Series,

Then

a well-established result obtained previously.

EXAMPLE 54

Use a series to evaluate

SOLUTION: We can use series (4) in Common Maclaurin Series, and write

EXAMPLE 55

EXAMPLE 56

Show how a series may be used to evaluate π.

SOLUTION: Since a series for tan⁻¹ x may prove helpful. Note that

and that a series for is obtainable easily by long division to yield

If we integrate this series term by term and then evaluate the definite integral, we get

(Compare with series (5) in Common Maclaurin Series and note especially that this series converges on −1 ≤ x ≤ 1.)

For x = 1 we have:

Here are some approximations for π using this series:

Since the series is alternating, the odd sums are greater, the even ones less, than the value of π. It is clear that several hundred terms may be required to get even two-place accuracy. There are series expressions for π that converge much more rapidly. (See Miscellaneous Free-Response Practice, Problem 12.)

EXAMPLE 57

Use a series to evaluate to four decimal places.

SOLUTION: Although cannot be expressed in terms of elementary functions, we can write a series for e^u, replace u by (−x²), and integrate term by term. Thus,

Since this is a convergent alternating series (with terms decreasing in magnitude and approaching 0), which will not affect the fourth decimal place. Then, correct to four decimal places,

^†C7. Power Series over Complex Numbers.

A complex number is one of the form a + bi, where a and b are real and i² = −1. If we allow complex numbers as replacements for x in power series, we obtain some interesting results.

Consider, for instance, the series

When x = yi, then (1) becomes

Then

since the series within the parentheses of equation (2) converge respectively to cos y and sin y. Equation (3) is called Euler’s formula. It follows from (3) that

e^{i π} = − 1,

and thus that

e^{i π} + 1 = 0,

sometimes referred to as Euler’s magic formula.

^† This is an optional topic not in the BC Course Description. We include it here because of the dramatic result.

Chapter Summary

In this chapter, we have reviewed an important BC Calculus topic, infinite series. We have looked at a variety of tests to determine whether a series converges or diverges. We have worked with functions defined as power series, reviewed how to derive Taylor series, and looked at the Maclaurin series expansions for many commonly used functions. Finally, we have reviewed how to find bounds on the errors that arise when series are used for approximations.