# 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)*