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Let A denote a complex unital Banach algebra. We characterize properties such as boundedness, relative compactness, and convergence of the sequence ${x^n(x-1)}_{n\in \mathbb{N}}$ for an arbitrary x ∈ A, using σ(x) and resolvent conditions. Under these circumstances, we investigate elements in the peripheral spectrum, and give further conclusions, also involving the behaviour of ${x^n}_{n\in \mathbb{N}}$ and ${1/n{\sum }_{k=0}^{n-1}{x}^k}_{n\in \mathbb{N}}$.