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Taylor Series
A function as an infinite sum of terms
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Taylor series 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! + .....
Examples
ex = 1 + x / 1! + x2 / 2! + x3/ 3! + ... (1)
ax = 1 + x ln a / 1! + (x ln a)2 / 2! + (x ln a)3 / 3! + .. (2)
ln x = 2( (x - 1) / (x + 1) + 1/3 ((x - 1) / (x + 1))3 + 1/5 ((x - 1) / (x + 1))5 + .. ) (3)
ln (1 + x) = x - x2 / 2 + x3/ 3 - x4 / 4 + x5/ 5 + ... (4)
sin x = x - x3 / 3! + x5/ 5! - x7 / 7! + ... (5)
cos x = 1 - x2 / 2! + x4/ 4! - x6 / 6! + ... (6)
tan x = x + 1/3 x3 + 2 /15 x5 + 17 / 315 x7 + ... (7)
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