Proposition: Convergence Behavior of the Sequence \((b^n)\)

The convergence behavior of the real sequence \((b^n)_{n\in\mathbb N}\) depends on the value of the real number \(b\). There are four possible cases:

\((1)\) For \(|b| < 1\) we have that \((b^n)_{n\in\mathbb N}\) is convergent with \(\lim_{n\to\infty} b^n=0\).

\((2)\) For \(b = 1\) we have \(\lim_{n\to\infty} b^n=1\).

\((3)\) For \(b = - 1\), the sequence \((b^n)_{n\in\mathbb N}\) is divergent. \((4)\) For \(|b| > 1\) we have that \((b^n)_{n\in\mathbb N}\) tends to infinity.

Proofs: 1

Proofs: 1 2

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