(related to Proposition: Convergence Behavior of the Sequence \((b^n)\))
There are four cases, how the real number \(b\) can influence the convergence of the real sequence \((b^n)_{n\in\mathbb N}\):
In this case, the sequence \((b^n)_{n\in\mathbb N}\) is convergent. This follows immediately from the corollary to the Archimedean Axiom about the existence of arbitrarilly small powers.
In this cases, it follows trivially that \(\lim_{n\to\infty} b^n=1\).
Because the set \((\mathbb R, + ,\cdot)\) of real numbers is a field, the set \((\mathbb R\setminus\{0\},\cdot)\) is an Abelian group. Therefore, it follows from the definition of the exponentiation for Abelian groups that
\[b^n := \begin{cases} 1 & \text{ if } n=0 \\ b\cdot b^{n-1} & \text{ if } n > 0 \end{cases}\]
It can be proven that \(-(-b)=b\) for all \(b\in\mathbb R\). Therefore, also \(-(-1)=(-1)(-1)=(-1)^2=1\). It follows that the real sequence \((-1)^n_{n\in\mathbb N}\) is alternating between the values \(1\) and \(-1\). It is simply the sequence \((1,-1,1,-1,1,-1,\ldots)\). Since this sequence neither converges to the number \(1\), nor it does converge to the number \(-1\), it is divergent, by definition.
In this case, the sequence \((b^n)_{n\in\mathbb N}\) tends to infinity. This follows immediately from the corollary to the Archimedean Axiom about the existence of powers exceeding positive real numbers.