INTRODUCTION

Much of the theory of complex sequences and series is analogous to that encountered in real calculus. In this section we explore the definitions of convergence and divergence for complex sequences and complex infinite series. In addition, we give some tests for convergence of infinite series. You are urged to pay special attention to what is said about geometric series since this type of series will be important in the later sections of this chapter.

Sequences

A sequence {z_n} is a function whose domain is the set of positive integers; in other words, to each integer n = 1, 2, 3, . . ., we assign a complex number z_n. For example, the sequence {1 + iⁿ} is

(1)

If lim_n→∞ z_n = L we say the sequence {z_n} is convergent. In other words, {z_n} converges to the number L if, for each positive number ϵ, an N can be found such that |z_n − L| < ϵ whenever n > N. As shown in FIGURE 19.1.1, when a sequence {z_n} converges to L, all but a finite number of the terms of the sequence are within every ϵ-neighborhood of L. The sequence {1 + iⁿ} illustrated in (1) is divergent, since the general term z_n = 1 + iⁿ does not approach a fixed complex number as n → ∞. Indeed, the first four terms of this sequence repeat endlessly as n increases.

EXAMPLE 1 A Convergent Sequence

The sequence converges, since

As we see from

and FIGURE 19.1.2, the terms of the sequence spiral toward the point z = 0. ≡

The following theorem should make intuitive sense.

EXAMPLE 2 Illustrating Theorem 19.1.1

The sequence converges to i. Note that Re(i) = 0 and Im(i) = 1. Then from

we see that Re(z_n) = 2n/(n² + 4) → 0 and Im(z_n) = n²/(n² + 4) → 1 as n → ∞. ≡

Series

An infinite series of complex numbers

is convergent if the sequence of partial sums {S_n}, where

S_n = z₁ + z₂ + z₃ + … + z_n,

converges. If S_n → L as n → ∞, we say that the sum of the series is L.

Geometric Series

For the geometric series

(2)

the nth term of the sequence of partial sums is

S_n = a + az + az² + … + azⁿ⁻¹. (3)

By multiplying S_n by z and subtracting this result from S_n, we obtain S_n − zS_n = a − azⁿ. Solving for S_n gives

(4)

Since zⁿ → 0 as n → ∞, whenever |z| < 1 we conclude from (4) that (2) converges to

when |z| < 1; the series diverges when |z| ≥ 1. The special geometric series

(5)

(6)

valid for |z| < 1, will be of particular usefulness in the next two sections. In addition, we shall use

(7)

in the alternative form

(8)

in the proofs of the two principal theorems of this chapter.

EXAMPLE 3 Convergent Geometric Series

The series

is a geometric series with a = (1 + 2i)/5 and z = (1 + 2i)/5. Since |z| = , the series converges and we can write

≡

An equivalent form of Theorem 19.1.2 is the familiar nth term test for divergence of an infinite series.

For example, the series (k + 5i)/k diverges since z_n = (n + 5i)/n → 1 as n → ∞. The geometric series (2) diverges when |z| ≥ 1, since, in this case, lim_n→∞ |zⁿ| does not exist.

EXAMPLE 4 Absolute Convergence

As in real calculus,

Thus in Example 4, because the series

converges absolutely, it is also convergent.

The following two tests are the complex versions of the ratio and root tests that are encountered in calculus:

We are interested primarily in applying these tests to power series.

Power Series

The notion of a power series is important in the study of analytic functions. An infinite series of the form

(11)

where the coefficients a_k are complex constants, is called a power series in z − z₀. The power series (11) is said to be centered at z₀, and the complex point z₀ is referred to as the center of the series. In (11), it is also convenient to define (z − z₀)⁰ = 1 even when z = z₀.

Circle of Convergence

Every complex power series has radius of convergence R, where R is a real number. Analogous to the concept of an interval of convergence in real calculus, when 0 < R < ∞, a complex power series (11) has a circle of convergence defined by |z − z₀| = R. The power series converges absolutely for all z satisfying |z − z₀| < R and diverges for |z − z₀| > R. See FIGURE 19.1.3. The radius R of convergence can be

1. zero (in which case (11) converges at only z = z₀),
2. a finite number (in which case (11) converges at all interior points of the circle |z − z₀| = R), or
3. ∞ (in which case (11) converges for all z).

A power series may converge at some, all, or none of the points on the actual circle of convergence.

EXAMPLE 5 Circle of Convergence

Consider the power series . By the ratio test (9),

Thus the series converges absolutely for |z| < 1. The circle of convergence is |z| = 1 and the radius of convergence is R = 1. Note that on the circle of convergence, the series does not converge absolutely, since the series of absolute values is the well-known divergent harmonic series (1/k). Bear in mind this does not say, however, that the series diverges on the circle of convergence. In fact, at z = −1, ((−1)^k+1/k) is the convergent alternating harmonic series. Indeed, it can be shown that the series converges at all points on the circle |z| = 1 except at z = 1. ≡

It should be clear from Theorem 19.1.4 and Example 5 that for a power series a_k(z − z₀)^k, the limit (9) depends on only the coefficients a_k. Thus, if

Similar remarks can be made for the root test (10) by utilizing .

EXAMPLE 6 Radius of Convergence

Consider the power series . Identifying a_n = (−1)^{n + 1}/n!, we have

Thus the radius of convergence is ∞; the power series with center 1 + i converges absolutely for all z. ≡

EXAMPLE 7 Radius of Convergence

Consider the power series . With a_n = , the root test in the form

shows that the radius of convergence of the series is R = . The circle of convergence is |z − 2i| = ; the series converges absolutely for |z − 2i| < . ≡

19.1 Exercises Answers to selected odd-numbered problems begin on page ANS-47.

In Problems 1–4, write out the first five terms of the given sequence.

1. {5iⁿ}
2. {2 + (−i)ⁿ}
3. {1 + e^nπi}
4. {(1 + i)ⁿ} [Hint: Write in polar form.]

In Problems 5–10, determine whether the given sequence converges or diverges.

5. 
6. 
7. 
8. 
9. 
10. {e^1/n + 2(tan⁻¹n)i}

In Problems 11 and 12, show that the given sequence {z_n} converges to a complex number L by computing lim_n→∞ Re(z_n) and lim_n→∞ Im(z_n).

In Problems 13 and 14, use the sequence of partial sums to show that the given series is convergent.

In Problems 15–20, determine whether the given geometric series is convergent or divergent. If convergent, find its sum.

In Problems 21–28, find the circle and radius of convergence of the given power series.

21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. Show that the power series is not absolutely convergent on its circle of convergence. Determine at least one point on the circle of convergence at which the power series converges.
30. 1. Show that the power series converges at every point on its circle of convergence.
    2. Show that the power series diverges at every point on its circle of convergence.