Category Archives: elementary

Does this product sequence converge?

Consider the following series: \begin{align*} a_1 &= \frac{4}{3}\\ a_2 &= \frac{4}{3}\frac{9}{8}\\ &\vdots\\ a_n &= \frac{4}{3}\frac{9}{8}\cdots\frac{(n+1)^2}{(n+1)^2-1} \end{align*} In other words, $a_n$ is the product of all the numbers of the form $n^2/(n^2 – 1)$ for $n=2,\dots, n+1$. Does $\lim_{n\to\infty} a_n$ exist?

The Smallest Number Paradox

The smallest number paradox goes like this: consider the natural numbers: 0,1,2,3,… . Each can be specified by a string of characters. For example, “0” itself specifies 0. However, on my computer there are only finitely many bits. Therefore, only finitely many numbers can be specified as a string on my computer. In other words, […]

Fibonacci sequence modulo m

The Fibonacci sequence is an infinite sequence of integers $f_0,f_1,f_2,\dots$ defined by the initial values $f_0 = f_1 = 1$ and the rule $$f_{n+1} = f_n + f_{n-1}$$ In other words, to get the next term you take the sum of the two previous terms. For example, it starts off with: $$1,1,2,3,5,8,13,21,34,55,\dots$$ You can define […]

What is a Liouville number?

An irrational number $r$ is called a Liouville number if for every positive integer $n$ there exists integers $p$ and $q \gt 1$ such that $$| r – p/q | \lt 1/q^n.$$ The requirement that $q \gt 1$ is crucial because otherwise, all irrational numbers would satisfy this definition. Liouville numbers are transcendental. In an […]

All set endomorphisms of a finite field are polynomial

Let $F$ be a finite field. Did you know that given any function $\varphi:F\to F$, there exists a polynomial $p\in F[x]$ such that $\varphi(a) = p(a)$ for all $a\in F$? It’s not hard to produce such the required polynomial: $$ p(x) = \sum_{a\in F} \left( \varphi(a)\prod_{b\not= a}(x – b)\prod_{b\not=a}(a-b)^{-1} \prod \right)$$ This works because every […]

Harmonic Numbers

The $n$th harmonic number $h(n)$ is defined as $$h(n) = \sum_{i=1}^n 1/i$$ The harmonic series is the associated series $\sum_{i=1}^\infty 1/i$ and it diverges. There are probably quite a few interesting ways to see this. My favourite is a simple comparison test: $$1/1 + 1/2 + 1/3 + \cdots\\ \geq 1/2 + 1/2 + 1/4 […]

Switching the order of summation

Characteristic functions have magical properties. For example, consider a double summation: $$\sum_{k=1}^M\sum_{r=1}^k a_{r,k}.$$ How do you switch the order of summation here? A geometric way to think of this is arrange the terms out and “see” that this sum must be equal to $$\sum_{r=1}^M\sum_{k=r}^M a_{r,k}.$$ I find this unsatisfactory because the whole point of good […]

When Is Squaring and Cubing a Group Homomorphism?

Let $G$ be a group. Define $\varphi:G\to G$ by $\varphi(x) = x^2$. When is $\varphi$ a homomorphism? Clearly, $\varphi$ is a homomorphism whenever $G$ is abelian. Conversely, if $\varphi$ is a homomorphism then for any $x,y\in G$, we get $xyxy = \varphi(xy) = \varphi(x)\varphi(y) = x^2y^2$. So, $xyxy = xxyy$. Canceling the $x$ from the […]

Graphing the Mandelbrot Set

A class of fractals known as Mandelbrot sets, named after Benoit Mandelbrot, have pervaded popular culture and are now controlling us. Well, perhaps not quite, but have you ever wondered how they are drawn? Here is an approximation of one: From now on, Mandelbrot set will refer to the following set: for any complex number […]