Let \(\sum_{k=0}^\infty a_k\) be an infinite complex series with \(a_k\neq 0\) for all \(k \ge K\), where \(K\in\mathbb N\) is some index (i.e. all but the first \(K\) sequence members \(a_k\) must be different from \(0\)). Furthermore, assume that there exists a real number \(\theta\) with \(0 < \theta < 1\) such that \[\left|\frac{a_{k+1}}{a_k}\right|\le \theta\text{ for all }k \ge K.\]
Then it follows that \(\sum_{k=0}^\infty a_k\) is an absolutely convergent complex series.
Proofs: 1
Proofs: 1