Sequence general term can be a(n) * x^n {power series}|, where a are rational coefficients, and n is number of terms. Power series converges if independent variable equals zero. x^n / n! converges at all x.
1 + a1 * u + a2 * u^2 / 2! + ... {associated series}, where ai are series coefficients.
Power series {expansion, function} {expansion, series} can replace function or integral.
e^x = 1 + x + x^2 / 2! + x^3 / 3! + ... {exponential series}.
sinh(x) = x + x^3 / 3! + x^5 / 5! + ... {hyperbolic expansion}. cosh(x) = 1 + x^2 / 2! + x^4 / 4! + ...
1 + a*b*x / c + (a * (a + 1) * b * (b + 1) * x^2) / (2 * c * (c + 1)) + ... {hypergeometric series} can converge at x. If x = 1, function equals (gamma(c) * gamma(c - a - b)) / (gamma(c - a) * gamma(c - b)).
1 + x + 2! * x^2 + 3! * x^3 + ... {Laguerre series}.
ln(1 + x) = x - x^2 / 2 + x^3 / 3 - x^4 / 4 + ... {logarithmic series}, where -1 <= x <+ +1. ln(N + 1) - ln(N) = 2 * (1 / (2*N - 1) + 1 / (3 * (2*N - 1)^3) + 1 / (5 * (2*N - 1)^5) + ...).
f(b) = f(0) + (sum from i = 0 to i = n - 1 of (function ith derivative at 0) / i!) * b^i + error term {Maclaurin expansion}. Maclaurin expansion is Taylor series with a = 0.
Fractions {radix fraction} can be sums of positive common fractions a/r + b / r^2 + c / r^3 + ..., where r is a number, and a b c are integers. Common fractions can be radix fractions, using any radix in any number system.
f(b) = f(a) + (sum from i = 0 to i = n - 1 of (function ith derivative at a) / i!) * (b - a)^i + error term {Taylor's theorem} {Taylor theorem} {Taylor series}. For two variables, Taylor series has twice as many terms {Taylor expansion}.
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Date Modified: 2022.0225