Proof: By Induction

(related to Proposition: Ratio Test For Absolutely Convergent Complex Series)

Note that changing a finite number of the sequence members of any infinite complex series \(\sum_{k=0}^\infty a_k\) does not change its convergence behavior. Therefore, by changing the first \(K\) (if any) sequence members \(a_k\), we can assume that

  1. \(a_k\neq 0\) for all \(k \in\mathbb N\) (i.e. all sequence members \(a_k\) are different from complex zero \(0\)),
  2. there exists a real number \(\theta\) with \(0 < \theta < 1\) such that \(\left|\frac{a_{k+1}}{a_k}\right|\le \theta\) for all \(k \in \mathbb N.\)

It follows from the proving principle by induction that (base case) \[|a_1|\le |a_0|\theta^0=|a_0|\cdot 1,\] and that (induction step), given \(|a_k|\le |a_0|\theta^{k-1},\) \[|a_{k+1}|\le |a_k|\cdot \theta\le |a_0|\theta^{k}.\]

From this result, it follows that the series \(\sum_{k=0}^\infty |a_0|\theta^{k}\) is a majorant of the series \(\sum_{k=0}^\infty a_k\). Due to the convergence of the infinite geometric series. \[\sum_{k=0}^\infty |a_0|\theta^{k}=|a_0|\sum_{k=0}^\infty \theta^{k}=\frac{|a_0|}{1-\theta}\] it follows from the majorant criterion for complex series that \(\sum_{k=0}^\infty a_k\) is an absolutely convergent complex series.

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