## Series

#### Arithmetic series

$t_n = t_1 + (n-1)d$
Where:
$$t_n$$ = the $$n^{th}$$ term
$$d$$ = the common difference, that is $$d = t_{n+1} - t_n$$

$\begin{eqnarray*} \sum_{i=1}^{n}t_i &=&t_1 + t_2 + t_3 + ... + t_n \\ &=& \frac{1}{2} n \left(t_1 + t_n \right) \end{eqnarray*}$

#### Geometric series

$t_n = t_1 \times r^{n-1}$
Where:
$$t_n$$ = the $$n^{th}$$ term
$$r$$ = the common ratio, that is $$r = \frac{t_{n+1}}{t_n}$$

$\begin{eqnarray*} \sum_{i=0}^{n}r^{i} &=&1 + r + r^2 + r^3 + ... + r^n \\ &=& \frac{1 - r^{n+1}}{1 - r} \end{eqnarray*}$

#### Taylor series

A Taylor series is a series expansion of a function about a point $$x_0$$

$\begin{eqnarray*} f(x) &=& \sum_{n=0}^{\infty} \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n \\ &=& f(x_0) + f^{'}(x_0)(x-x_0) + \frac{f^{' '}(x_0)}{2!}(x-x_0)^2 + \frac{f^{' ' '}(x_0)}{3!}(x-x_0)^3 + ...\ \end{eqnarray*}$

#### Maclaurin series

A Maclaurin series is a series expansion of a function about a point $$0$$

$\begin{eqnarray*} f(x) &=& \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}(x)^n \\ &=& f(0) + f^{'}(0)x + \frac{f^{' '}(0)}{2!}x^2 + \frac{f^{' ' '}(x)}{3!}x^3 + ...\ \end{eqnarray*}$

#### Fourier series

The Fourier series of a function $$f(x)$$ over $$[-L, L]$$
$f(x) = \frac{1}{2} a_0 + \sum_{n=1}^{\infty} \left[ a_n \cos \left(\frac{n \pi x}{L} \right) + b_n \sin \left(\frac{n \pi x}{L} \right) \right]$
Where:

$a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos \left(\frac{n \pi x}{L} \right)$ $b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin \left(\frac{n \pi x}{L} \right)$

The Fourier series of a function $$f(x)$$ over $$[-\pi, \pi]$$
$f(x) = \frac{1}{2} a_0 + \sum_{n=1}^{\infty} \left[ a_n \cos (n x) + b_n \sin (n x) \right]$

Where:

$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos (n x)$ $b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin (n x)$

Orthogonality of trigonometric functions
For all integers $$m$$ and $$n$$
$\int_{-L}^{L} \cos \left(\frac{n \pi x}{L} \right) \sin \left(\frac{m \pi x}{L} \right) dx = 0$

If $$m \ne n$$
$\int_{-L}^{L} \cos \left(\frac{n \pi x}{L} \right) \cos \left(\frac{m \pi x}{L} \right) dx = 0$ $\int_{-L}^{L} \sin \left(\frac{n \pi x}{L} \right) \sin \left(\frac{m \pi x}{L} \right) dx = 0$

If $$n \ne 0$$
$\int_{-L}^{L} \cos^2 \left(\frac{n \pi x}{L} \right) dx = L$ $\int_{-L}^{L} \sin^2 \left(\frac{n \pi x}{L} \right) dx = L$