130 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER SERIES
Definition 4.5: Cauchy product
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Let L an and L bn be convergent series, where an and bn are complex num-
n=O n=O
bers. The Cauchy product of the two series given above is defined to be the
oo n
series L en, where Cn = L akbn-k·
n=O k=O
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t Corollary 4.1 If L lznl converges, then L Zn converges. In other words,
n=l n=O
absolute convergence implies convergence for complex series as well as for real
series.
Proof. The proof is left as an exercise.
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