List of some basic formulas related to approximate calculations using series:
- 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 ). - 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
] - Geometric progression:
[
S = a + ar + ar^2 + ar^3 + \ldots = \frac{a}{1 – r} \quad (|r| < 1)
] - 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)
] - Benford series (for logarithmic approximations):
[
P(d) = \log_{10}(d + 1) – \log_{10}(d) \quad (d = 1, 2, \ldots, 9)
] - 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.