Geometric Series

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math

A number series where the value of the next entry is equal to the current entry multiplied by some constant \( r \).

For such a series we can use the following formulae to find the \(n^{\text{nth}}\) element, the sum from the first to the \(n^{\text{nth}}\) element and the sum to infinity (what the sum of the series tends to as \( n \) approaches infinity).

\begin{align} u_n &= ar^{n--1} \
S_n &= \frac{a(1 - r^n)}{1-r} \
S_{\infty} &= \frac{a}{1-r}, \quad for |r| < 1 \end{align}