List of some formulas related to series of increased complexity:

  1. 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.
  2. 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 ).
  3. 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 .
  4. Laurent series:
    [
    f(z) = \sum_{n=-\infty}^{\infty} a_n z^n
    ]
    where ( a_n ) are the coefficients of the Laurent series.
  5. 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.
  6. Maclaurin series (a special case of Taylor series):
    [
    f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n
    ]

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