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Proposition: Direct Comparison Test For Absolutely Convergent Complex Series

In order to test, if a complex series \(\sum_{k=0}^\infty x_k\) is an absolutely convergent complex series, it suffices to find a convergent real series \(\sum_{k=0}^\infty y_k\) with \(|x_k|\le y_k\) for all \(k\). Such a series \(\sum_{k=0}^\infty y_k\) is called the majorant of the series \(\sum_{k=0}^\infty x_k\).

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Proofs: 1 2

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References

Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983