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Binomial Theorem
----------------

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

Exponential Function
--------------------

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

Cauchy-Schwarz Inequality
-------------------------

$$
  \left( \sum_{k=1}^n a_k b_k \right)^2 \leq
  \left( \sum_{k=1}^n a_k^2 \right)
  \left( \sum_{k=1}^n b_k^2 \right).
$$

Bayes' Theorem
--------------

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

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{align}
  \sum_{y < n \le x} f(n)
    = & \int_y^x f(t) \, dt + \int_y^x (t - [t]) f'(t) \, dt
       \notag \\
       & + f(x)([x] - x) - f(y)([y] - y). \label{theorem}
\end{align}

_Proof._ Let $ m = [y] $ and $ k = [x]. $ For
integers $ n $ and $ n - 1 $ in $ [y, x] $ we have
\begin{align*}
  \int_{n - 1}^n [t] f'(t) \, dt
    & = \int_{n - 1}^n (n - 1) f'(t) \, dt \\
    & = (n - 1) \{ f(n) - f(n - 1) \} \\
    & = \{n f(n) - (n - 1) f(n - 1)\} - f(n).
\end{align*}
Summing from $ n = m + 2 $ to $ n = k $ we find
that the first sum telescopes, hence
\begin{align*}
  \int_{m + 1}^k [t] f'(t) \, dt
    & = 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 \le x} f(n).
\end{align*}
Therefore
\begin{align}
  \sum_{y < n \le x} f(n)
    & = - \int_{m + 1}^k [t] f'(t) \, dt + k f(k) - m f(m + 1) \notag \\
    & = - \int_y^x [t] f'(t) \, dt + k f(x) - m f(y). \label{summation}
\end{align}
Integration by parts gives us
$$
\int_y^x f(t) \, dt = x f(x) - y f(y) - \int_y^x t f'(t) \, dt,
$$
and when this is combined with \eqref{summation}
we obtain \eqref{theorem}.

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.

Table of Number Theory Functions
--------------------------------

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

About This Demo
---------------

This is a demo of a MathB.in post written in
Markdown + LaTeX.

2024-10-03 10:50:44