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
