The sum of a geometric series is all you need! – Machine Learning Research Blog
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fairly easy
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I sometimes joke with my students about one of the main tools I have been using in the last ten years: the explicit sum of a geometric series. Why is this?

From numbers to operators

The simplest version of this basic result for real numbers is the following: $$ \forall r

eq 1, \ \forall n \geq 0, \ \sum_{k=0}^n r^k = \frac{1-r^{n+1}}{1-r},$$ and is typically proved by multiplying the two sides by \(1-r\) and forming a telescoping sum. When \(|r|<1\), we can let \(n\) tend to infinity and get $$ \forall |r| < 1, \ \sum_{k=0}^\infty r^k = \frac{1}{1-r}.$$

Proofs without words. There is a number of classical proofs of the last identities, many of them without words, presented in the beautiful series of books by Roger Nelsen [1, 2, 3]. I particularly like the two below.

Proof of the infinite sum of a geometric series with \(r=\frac{1}{2}.\) The area of the right triangle which is the half of a square with side length equal to \(2\), is equal to \(2\) and to the sum of the areas of the smaller triangles, that is, \(2 = \frac{1}{1- \frac{1}{2}}= 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots\). Adapted from [3, p. 155].

Proof of the finite sum of a geometric series, started at \(k=0\) up to \(k=n=5\). The area of the full square of unit side length is equal to the sum of the areas of all yellow rectangles plus the pink one, that is, \(1 = (1-r) \sum_{k=0}^n r^k + r^{n+1}\). Adapted from [1, p. 118].

High school: from philosophy to trigonometry. Before looking at extensions beyond real numbers, I can't resist mentioning some of Zeno's paradoxes, like Achilles and the tortoise, which are intimately linked with the sum of a geometric series (and which were a highlight of my high school philosophy "career").

Speaking of high school, it is interesting to note that there, the core identity of this blog post, is often used as \(a^n-b^n = (a-b)(a^n+a^{n-1}b+\cdots+ab^{n-2}+b^{n-1})\) in order to factorize polynomials (and not in the other way around like done…
Francis Bach, Ahmed Mazari
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