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# 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)

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## Citation

• The Engineering ToolBox (2011). Taylor Series. [online] Available at: https://www.engineeringtoolbox.com/taylor-series-d_1750.html [Accessed Day Month Year].

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9.19.12