(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
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.
