05-math.Rmd

# Useful Math Facts {#math}

## Exponents and Logarithms
for positive constants $a$ and $b$:

$a^0 = 1$

$a^{-n}=\frac{1}{a^n}$ for positive integer $n$

$a^{x+y}=a^x a^y$

$a^{x-y}=\frac{a^x}{a^y}$

$(a^x)^y=a^{xy}$

$(ab)^x=a^xb^x$


$\log_a(x) = y$ implies $a^{y} = x$

If you're working in statistics it's usually a safe bet that $\log(x)$ means $\log_e(x)$ not $\log_{10}(x)$.

$\log(1) = 0$

$\log(e) = 1$

$\log(xy)=\log(x) + \log(y)$

$\log(x/y)=\log(x)-\log(y)$

## Sequences and Series

### Monotonicity

A sequence of numbers $\{x_1,x_2,\ldots\}$ is *monotone increasing* if $x_n \le x_{n+1}$ for all $n$. If $x_n \ge x_{n+1}$ for all $n$, the sequence is *monotone decreasing*

### Geometric series

$\sum\limits_{k=0}^{n} a^k =\frac{1- a^{n+1}}{1-a}$

for $|a|<1$ $\sum\limits_{k=0}^{\infty} a^k =\frac{1}{1-a}$

### Taylor series

$e^x=\sum\limits_{n=0}^{\infty}\frac{x^n}{n!}$

### Binomial Theorem

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

## Calculus - Derivatives

### Maxima and Minima

## Calculus - Integrals

## Miscellaneous

### Gamma function

The Gamma function is defined as $\Gamma(\alpha) =\int\limits_{0}^{\infty}x^{\alpha-1}e^{-x}\, dx$ for $\alpha > 0$

$\Gamma(1)=1$

$\Gamma(\alpha)=(\alpha-1)\Gamma(\alpha-1)$

$\alpha\Gamma(\alpha)=\Gamma(\alpha+1)$

$\Gamma(n)=(n-1)!$ for integer $n>0$