Geometric Sequence

analysis

Definition

Geometric Sequence

A sequence is called geometric if the ratio between two consecutive terms remains constant:

$$\forall n\in \mathcal{D}:q=\frac{a_{n+1}}{a_n}$$

The sequence would be $(a_0,a_0\cdot q,a_0\cdot q^2,\dots )$.

An alternative representation:

$$\begin{array}{r}{a}_n=a_0\cdot q^n\end{array}$$

Convergence

A geometric sequence $a_n=a_0\cdot q^n$ is convergent for $q>-1$:

$$\underset{n\to \infty }{\lim }=\{\begin{array}{ll}0& |q|<1\\ 1& q=1\\ \infty & q>1\end{array}$$

and divergent for $q\le -1$.