(I am launching a new series called “Epsilons”, where I explain a single concept or tool in 500 words or less. In mathematics, the greek epsilon (ε) denotes an infinitesimal quantity. Thus, what ε is for mathematicians is what a *bit* is for computer scientists.)

One of my favorite formulas is the closed-form of the geometric series.

I am amazed by its ubiquity: whether we are solving basic problems or pushing the boundaries of science, the geometric series often makes a surprising appearance.

Here is how to derive it from first principles.

Let’s start with the basics: like any other series, the geometric series is the limit of its partial sums.

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