List of some formulas related to series of increased complexity:
- Geometric series:
[
S_n = a \frac{1 – r^n}{1 – r}, \quad (r \neq 1)
]
where ( S_n ) is the sum of the first ( n ) terms, ( a ) is the first term, ( r ) is the denominator. - Taylor series:
[
f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x – a)^n
]
where ( f^{(n)}(a) ) is the ( n )-th derivative of the function ( f ) at the point ( a ). - Fourier series:
[
f(x) = a_0 + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx))
]
where ( a_n ) and ( b_n ) are the Fourier coefficients . - Laurent series:
[
f(z) = \sum_{n=-\infty}^{\infty} a_n z^n
]
where ( a_n ) are the coefficients of the Laurent series. - Sum of an infinite geometric series:
[
S = \frac{a}{1 – r}, \quad (|r| < 1)
]
where ( S ) is the sum of the series, ( a ) is the first term, ( r ) is the denominator. - Maclaurin series (a special case of Taylor series):
[
f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n
]