Calculus AB and Calculus BC

CHAPTER 10 Sequences and Series

B. INFINITE SERIES

B1. Definitions.

Infinite series

If an is a sequence of real numbers, then an infinite series is an expression of the form

Image

The elements in the sum are called terms; an is the nth or general term of the series.

EXAMPLE 7

A series of the form Image is called a p-series.

The p-series for p = 2 is Image

EXAMPLE 8

The p-series with p = 1 is called the harmonic series:

Image

EXAMPLE 9

A geometric series has a first term, a, and common ratio of terms, r:

Image

If there is a finite number S such that

Image

then we say that infinite series is convergent, or converges to S, or has the sum S, and we write, in this case,

Image

When there is no source of confusion, the infinite series (1) may be indicated simply by

Image

EXAMPLE 10

Show that the geometric series Image converges to 2.

SOLUTION: Let S represent the sum of the series; then:

Image

EXAMPLE 11

Show that the harmonic series Image diverges.

SOLUTION The terms in the series can be grouped as follows:

Image

This sum clearly exceeds

Image

which equals

Image

Since that sum is not bounded, it follows that Image diverges to ∞.

B2. Theorems About Convergence or Divergence of Infinite Series.

The following theorems are important.

THEOREM 2a. If Image converges, then Image

This provides a convenient and useful test for divergence, since it is equivalent to the statement: If an does not approach zero, then the series Image diverges. Note, however, particularly that the converse of Theorem 2a is not true. The condition that an approach zero is necessary but not sufficient for the convergence of the series. The harmonic series Image is an excellent example of a series whose nth term goes to zero but that diverges (see Example 11 above). The series Image diverges because Image not zero; the series Image does not converge (as will be shown shortly) even though Image

THEOREM 2b. A finite number of terms may be added to or deleted from a series without affecting its convergence or divergence; thus

Image

(where m is any positive integer) both converge or both diverge. (Note that the sums most likely will differ.)

THEOREM 2c. The terms of a series may be multiplied by a nonzero constant without affecting the convergence or divergence; thus

Image

both converge or both diverge. (Again, the sums will usually differ.)

THEOREM 2d. If Image both converge, so does Image

THEOREM 2e. If the terms of a convergent series are regrouped, the new series converges.

B3. Tests for Convergence of Infinite Series.

THE nth TERM TEST

If Image diverges.

NOTE: When working with series, it’s a good idea to start by checking the nth Term Test. If the terms don’t approach 0, the series cannot converge. This is often the quickest and easiest way to identify a divergent series.

(Because this is the contrapositive of Theorem 2a, it’s always true. But beware of the converse! Seeing that the terms do approach 0 does not guarantee that the series must converge. It just means that you need to try other tests.)

EXAMPLE 12

Does Image converge or diverge?

SOLUTION: Since Image the series Image diverges by the nth Term Test.

THE GEOMETRIC SERIES TEST

A geometric series Image converges if and only if |r| < 1.

If |r| < 1, the sum is Image

The series cannot converge unless it passes the nth Term Test; Image only if |r| < 1. As noted earlier, this is a necessary condition for convergence, but may not be sufficient. We now examine the sum using the same technique we employed in Example 10:

Image

EXAMPLE 13

Does 0.3 + 0.03 + 0.003 + · · · converge or diverge?

SOLUTION: The series 0.3 + 0.03 + 0.003 + · · · is geometric with a = 0.3 and r = 0.1. Since |r| < 1, the series converges, and its sum is

Image

NOTE: Image = 0.333 …, which is the given series.

B4. Tests for Convergence of Nonnegative Series.

The series Image is called a nonnegative series if an ≥ 0 for all n.

THE INTEGRAL TEST

If f (x) is a continuous, positive, decreasing function and f (n) = an, then Image converges if and only if the improper integral Image converges.

EXAMPLE 14

Does Image converge?

SOLUTION: The associated improper integral is

Image

which equals

Image

The improper integral and the infinite series both diverge.

EXAMPLE 15

Test the series Image for convergence.

SOLUTION: Image

by an application of L’Hôpital’s Rule. Thus Image converges.

THE p-SERIES TEST

A p-series Image converges if p > 1, but diverges if p ≤ 1.

This follows immediately from the Integral Test and the behavior of improper integrals of the form Image

EXAMPLE 16

Does the series Image converge or diverge?

SOLUTION: The series Image is a p-series with p = 3;

hence the series converges by the p-Series Test.

EXAMPLE 17

Does the series Image converge or diverge?

SOLUTION: Image diverges, because it is a p-series with Image

THE COMPARISON TEST

We compare the general term of Image the nonnegative series we are investigating, with the general term of a series known to converge or diverge.

(1) If Image converges and an Image un, then Image converges.

(2) If Image diverges and an Image un, then Image diverges.

Any known series can be used for comparison. Particularly useful are p-series, which converge if p > 1 but diverge if p Image 1, and geometric series, which converge if |r| < 1 but diverge if |r| Image 1.

EXAMPLE 18

Does Image converge or diverge?

SOLUTION: Since Image and the p-series Image converges, Image converges by the Comparison Test.

EXAMPLE 19

Does the series Image converge or diverge?

SOLUTION: Image diverges, since

Image

the latter is the general term of the divergent p-series Image where Image and Image

Remember in using the Comparison Test that you may either discard a finite number of terms or multiply each term by a nonzero constant without affecting the convergence of the series you are testing.

EXAMPLE 20

Show that Image converges.

SOLUTION: For Image is a convergent geometric series with Image

THE LIMIT COMPARISON TEST

If Image is finite and nonzero, then Image and Image both converge or both diverge.

This test is useful when the direct comparisons required by the Comparison Test are difficult to establish. Note that, if the limit is zero or infinite, the test is inconclusive and some other approach must be used.

EXAMPLE 21

Does Image converge or diverge?

SOLUTION: This series seems to be related to the divergent harmonic series, but Image so the comparison fails. However, the Limit Comparison Test yields:

Image

Since Image also diverges by the Limit Comparison Test.

THE RATIO TEST

Let Image if it exists. Then Image converges if L < 1 and diverges if L > 1.

If L = 1, this test is inconclusive; apply one of the other tests.

EXAMPLE 22

Does Image converge or diverge?

SOLUTION: Image

Therefore this series converges by the Ratio Test.

EXAMPLE 23

Does Image converge or diverge?

SOLUTION: Image

and

Image

(See §E2.) Since e > 1, Image diverges by the Ratio Test.

EXAMPLE 24

If the Ratio Test is applied to any p-series, Image then

Image

But if p > 1 then Image converges, while if p Image 1 then Image diverges. This illustrates the failure of the Ratio Test to resolve the question of convergence when the limit of the ratio is 1.

THE nth ROOT TEST

Let Image if it exists. Then Image converges if L < 1 and diverges if L > 1.

If L = 1 this test is inconclusive; try one of the other tests.

Note that the decision rule for this test is the same as that for the Ratio Test.

EXAMPLE 25

The series Image converges by the nth Root Test, since

Image

B5. Alternating Series and Absolute Convergence.

Any test that can be applied to a nonnegative series can be used for a series all of whose terms are negative. We consider here only one type of series with mixed signs, the so-called alternating series. This has the form:

Image

where ak > 0. The series

Image

is the alternating harmonic series.

THE ALTERNATING SERIES TEST

An alternating series converges if:

(1) an + 1 < an for all n, and

(2) Image

EXAMPLE 26

Does the series Image converge or diverge?

SOLUTION: The alternating harmonic series Image converges, since

(1) Image for all n and

(2) Image

EXAMPLE 27

Does the series Image converge or diverge?

SOLUTION: The series Image diverges, since we see that Image is 1, not 0. (By the nth Term Test, if an does not approach 0, then Image does not converge.)

DEFINITION

Absolute convergence

A series with mixed signs is said to converge absolutely (or to be absolutely convergent) if the series obtained by taking the absolute values of its terms converges; that is, Image converges absolutely if Image converges.

A series that converges but not absolutely is said to converge conditionally (or to be conditionally convergent). The alternating harmonic series converges conditionally since it converges, but does not converge absolutely. (The harmonic series diverges.)

When asked to determine whether an alternating series is absolutely convergent, conditionally convergent, or divergent, it is often advisable to first consider the series of absolute values. Check first for divergence, using the nth Term Test. If that test shows that the series may converge, investigate further, using the tests for nonnegative series. If you find that the series of absolute values converges, then the alternating series is absolutely convergent. If, however, you find that the series of absolute values diverges, then you’ll need to use the Alternating Series Test to see whether the series is conditionally convergent.

EXAMPLE 28

Determine whether Image converges absolutely, converges conditionally, or diverges.

SOLUTION: We see that Image not 0, so by the nth Term Test the series Image is divergent.

EXAMPLE 29

Determine whether Image converges absolutely, converges conditionally, or diverges.

SOLUTION: Note that, since Image the series passes the nth Term Test.

Image

But Image is the general term of a convergent p-series (p = 2), so by the Comparison Test the nonnegative series converges, and therefore the alternating series converges absolutely.

EXAMPLE 30

Determine whether Image converges absolutely, converges conditionally, or diverges.

SOLUTION: Image is a p-series with Image so the nonnegative series diverges.

We see that Image

so the alternating series converges; hence Image is conditionally convergent.

APPROXIMATING THE LIMIT OF AN ALTERNATING SERIES

Evaluating the sum of the first n terms of an alternating series, given by Image yields an approximation of the limit, L. The error (the difference between the approximation and the true limit) is called the remainder after n terms and is denoted by Rn. When an alternating series is first shown to pass the Alternating Series Test, it’s easy to place an upper bound on this remainder. Because the terms alternate in sign and become progressively smaller in magnitude, an alternating series converges on its limit by oscillation, as shown in Figure N10–1.

Image

FIGURE N10–1

Error bound

Because carrying out the approximation one more term would once more carry us beyond L, we see that the error is always less than that next term. Since |Rn | < an + 1, the error bound for an alternating series is the first term omitted or dropped.

EXAMPLE 31

The series Image passes the Alternating Series Test; hence its sum differs from the sum

Image

by less than Image which is the error bound.

EXAMPLE 32

How many terms must be summed to approximate to three decimal places the value of Image

SOLUTION: Since Image the series converges by the Alternating Series Test; therefore after summing a number of terms the remainder (error) will be less than the first omitted term.

We seek n such that Image Thus n must satisfy (n + 1)2 > 1000, or n > 30.623. Therefore 31 terms are needed for the desired accuracy.