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Binomial Theorem

$(x + y)^{n} = \sum_{k = 0}^{n} (\binom{n}{k}) x^{n - k} y^{k} .$

Exponential Function

$e^{x} = lim_{n \to \infty} {(1 + \frac{x}{n})}^{n} .$

Cauchy-Schwarz Inequality

${(\sum_{k = 1}^{n} a_{k} b_{k})}^{2} \leq (\sum_{k = 1}^{n} a_{k}^{2}) (\sum_{k = 1}^{n} b_{k}^{2}) .$

Bayes' Theorem

$P (A ∣ B) = \frac{P (B ∣ A) P (A)}{P (B)} .$

Euler's Summation Formula

Theorem (Euler's summation formula). If $f$ has a continuous derivative $f^{'}$ on the interval $[y, x],$ where $0 < y < x,$ then $\begin{aligned} \sum_{y < n \leq x} f (n) = & \int_{y}^{x} f (t) d t + \int_{y}^{x} (t - [t]) f^{'} (t) d t \\ (1) & + f (x) ([x] - x) - f (y) ([y] - y) . \end{aligned}$

Proof. Let $m = [y]$ , $k = [x] .$ For integers $n$ and $n - 1$ in $[y, x]$ we have $\begin{aligned} \int_{n - 1}^{n} [t] f^{'} (t) d t & = \int_{n - 1}^{n} (n - 1) f^{'} (t) d t \\ = (n - 1) {f (n) - f (n - 1)} \\ = {n f (n) - (n - 1) f (n - 1)} - f (n) . \end{aligned}$ Summing from $n = m + 2$ to $n = k$ we find the first sum telescopes, hence $\begin{aligned} \int_{m + 1}^{k} [t] f^{'} (t) d t & = k f (k) - (m + 1) f (m + 1) - \sum_{n = m + 2}^{k} f (n) \\ = k f (k) - m f (m + 1) - \sum_{y < n \leq x} f (n) . \end{aligned}$ Therefore $\begin{aligned} \sum_{y < n \leq x} f (n) & = - \int_{m + 1}^{k} [t] f^{'} (t) d t + k f (k) - m f (m + 1) \\ (2) & = - \int_{y}^{x} [t] f^{'} (t) d t + k f (x) - m f (y) . \end{aligned}$ Integration by parts gives us $\int_{y}^{x} f (t) d t = x f (x) - y f (y) - \int_{y}^{x} t f^{'} (t) d t,$ and when this is combined with $(2)$ we obtain $(1)$ . ∎

Hello World Program

Here is an example of "hello, world" program written in the C programming language:

#include <stdio.h>

int main()
{
    printf("hello, world\n");
    return 0;
}

Issac Newton Quotes

Issac Newton was relatively modest about his achievements, writing in a letter to Robert Hooke in February 1676:

If I have seen further it is by standing on the shoulders of giants.

In a later memoir, Newton wrote:

I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.

To read more about Newton, see the Wikipedia entry on Issac Newton.

Table of Number Theory Functions

The following table shows information about a few important functions in number theory.

Name	Notation	First few values	Multiplicative property
Möbius function	$μ (n)$	$1, - 1, - 1, 0, - 1$	Multiplicative
Euler's totient function	$φ (n)$	$1, 1, 2, 2, 4$	Multiplicative
Mangoldt function	$Λ (n)$	$0, \log 2, \log 3, \log 2, \log 5$	Not multiplicative
Liouville's function	$λ (n)$	$1, - 1, - 1, 1, - 1$	Completely multiplicative

About This Demo

This is a demo of a self-rendering Markdown + LaTeX document rendered with TeXMe. To learn more about what TeXMe is and how to use it, visit github.com/susam/texme.