Epsilons, no. 1: The geometric series

(One of) the building blocks of mathematics

(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.)

The geometric series

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.

The geometric series as the limit of its partial sums

Our task is to find that limit.

There is an issue: the number of terms depend on N. Thus, we can’t take the limit term by term.

The trick is to notice that multiplying the partial sums by (-q) yields a polynomial that can be used to eliminate all but two terms.

Adding them together yields a simple and manageable expression for the partial sums.

I know, this feels like pulling out a rabbit from a hat. Trust me, after you have seen this trick a few times, it’ll feel like second nature.

The result is called a telescopic sum. Thus, the partial sums are significantly simpler now.

The partial sums, simplified

We are almost done. Before we study the limit of partial sums, let’s focus on qᴺ.

Its limiting behavior (as N goes to ∞) is quite simple:

The limits of qᴺ

With this, we are ready to put all pieces together. The geometric series is convergent for all |q| < 1, with a nice and simple closed-form expression as the cherry on top.

The closed-form expression of the geometric series

This can be beautifully visualized in the case of q = 1/2.

(Almost) the geometric series for q = 1/2

Where does the geometric series appear? For instance, when deriving a closed-form expression for the Fibonacci numbers. Or, tossing coins ad infinitum. There are countless applications.

This simple formula is one of the building blocks of mathematics, and it should be under the belt of anyone who is interested in looking behind the curtain of science, engineering, and mathematics.