On the differences of the consecutive powers of Banach algebra elements

Let A denote a complex unital Banach algebra. We characterize properties such as boundedness, relative compactness, and convergence of the sequence ${\displaystyle {\left\{x^n(x-1)\right\}}_{n\in \mathbb{N}}}$ 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 ${\displaystyle {\left\{x^n\right\}}_{n\in \mathbb{N}}}$ and ${\displaystyle {\left\{\frac{1}{n}{\sum }_{k=0}^{n-1}{x}^k\right\}}_{n\in \mathbb{N}}}$.