Taylor Series
Function as an infinite sum of terms.
A Taylor serie is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. The generic expression:
f(x) = f(a) + f'(a) (x - a) / 1! + f''(a) (x - a)2 / 2! + ..... (1)
Examples
ex = 1 + x / 1! + x2 / 2! + x3 / 3! + ... (2)
ax = 1 + x ln(a) / 1! + (x ln(a))2 / 2! + (x ln(a))3 / 3! + .. (3)
ln(x) = 2( (x - 1) / (x + 1) + 1/3 ((x - 1) / (x + 1))3+ 1/5 ((x - 1) / (x + 1))5+ ..) (4)
ln(1 + x) = x - x2 / 2 + x3 / 3 - x4 / 4 + x5 / 5 + ... (5)
sin(x) = x - x3 / 3! + x5 / 5! - x7 / 7!+ ... (6)
cos(x) = 1 - x2 / 2! + x4 / 4! - x6 / 6!+ ... (7)
tan(x) = x + 1/3 x3 + 2 /15 x5 + 17 / 315 x7 + ... (8)
