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Maths formulas

Series

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

\[\begin{eqnarray*} \sum_{i=1}^{n}t_i &=&t_1 + t_2 + t_3 + ... + t_n \\ &=& \frac{1}{2} n \left(t_1 + t_n \right) \end{eqnarray*}\]
\[t_n = t_1 \times r^{n-1} \]
Where:
\(t_n\) = the \( n^{th}\) term
\(r\) = the common ratio, that is \(r = \frac{t_{n+1}}{t_n}\)

\[\begin{eqnarray*} \sum_{i=0}^{n}r^{i} &=&1 + r + r^2 + r^3 + ... + r^n \\ &=& \frac{1 - r^{n+1}}{1 - r} \end{eqnarray*}\]
A Taylor series is a series expansion of a function about a point \(x_0\)

\[\begin{eqnarray*} f(x) &=& \sum_{n=0}^{\infty} \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n \\ &=& f(x_0) + f^{'}(x_0)(x-x_0) + \frac{f^{' '}(x_0)}{2!}(x-x_0)^2 + \frac{f^{' ' '}(x_0)}{3!}(x-x_0)^3 + ...\ \end{eqnarray*}\]
A Maclaurin series is a series expansion of a function about a point \(0\)

\[\begin{eqnarray*} f(x) &=& \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}(x)^n \\ &=& f(0) + f^{'}(0)x + \frac{f^{' '}(0)}{2!}x^2 + \frac{f^{' ' '}(x)}{3!}x^3 + ...\ \end{eqnarray*}\]
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 \(n \ne 0\)
\[\int_{-L}^{L} \cos^2 \left(\frac{n \pi x}{L} \right) dx = L\] \[\int_{-L}^{L} \sin^2 \left(\frac{n \pi x}{L} \right) dx = L\]