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
e^{x} = 1 + x / 1! + x^{2} / 2! + x^{3}/ 3! + ... (2)
a^{x} = 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  x^{2} / 2 + x^{3}/ 3  x^{4} / 4 + x^{5}/ 5 + ... (5)
sin x = x  x^{3} / 3! + x^{5}/ 5!  x^{7} / 7!+ ... (6)
cos x = 1  x^{2} / 2! + x^{4}/ 4!  x^{6} / 6!+ ... (7)
tan x = x + 1/3 x^{3} + 2 /15 x^{5} + 17 / 315 x^{7} + ... (8)
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