List of some basic formulas related to approximate calculations using series:

  1. Taylor series:
    [
    f(x) = f(a) + f'(a)(x – a) + \frac{f”(a)}{2!}(x – a)^2 + \frac{f”'(a)}{3!}(x – a)^3 + \ldots
    ]
    where ( f^{(n)}(a) ) is the n-th derivative of the function ( f ) at the point ( a ).
  2. Maclaurin series (a special case of Taylor series for (a = 0)):
    [
    f(x) = f(0) + f'(0)x + \frac{f”(0)}{2!}x^2 + \frac{f”'(0)}{3!}x^3 + \ldots
    ]
  3. Geometric progression:
    [
    S = a + ar + ar^2 + ar^3 + \ldots = \frac{a}{1 – r} \quad (|r| < 1)
    ]
  4. Fourier series (for periodic functions):
    [
    f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos\left(\frac{2\pi nx}{T}\right ) + b_n \sin\left(\frac{2\pi nx}{T}\right) \right)
    ]
  5. Benford series (for logarithmic approximations):
    [
    P(d) = \log_{10}(d + 1) – \log_{10}(d) \quad (d = 1, 2, \ldots, 9)
    ]
  6. Laurent series (for analytic functions):
    [
    f(z) = \sum_{n=-\infty}^{\infty} a_n (z – z_0)^n
    ]

These formulas can be used for various approximate calculations in mathematics and cryptanalysis.

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