128 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER SERIES
Definition 4.4: Infinite series
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The formal expression L zk = z1 + z 2 + · · · + Zn + · · · is called an infinite
k = l
series, and z 1 , Z2, ... are called the terms of the series.
n
If there is a complex number S for which S = lim Sn = lim L Zk> we
n.-oo n~oo k=l
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say that the infinite series L Zk converges to S and that S is the sum of the
k = l
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infinite series. V\' hen convergence occurs, we write S = L Zk ·
k = I
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T he series L zk is said to be absolutely convergent provided that the
k= l
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(real) series of magnitudes L lzkl converges.
k=I
If a series does not converge, we say that it dive r ges.
Remark 4.2 The first finitely many tenns of a series do not affect its con-
vergence or divergence and, in t his respect, the beginning index of a series is
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irrelevant. Thus, we will conclude that if a series L zk converges, then so
k=N+l
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does L Zk, where z1, z2, ... , ZN is any finite collection of terms. A similar
k=l
remark applies to determining divergence of a series. •
As you might expect, many of the results concerning real series carry over to
complex series. We now give several of the more standard theorems for complex
series, along with examples of how they are used.
