# Taylor Series

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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