Main Concept

The Taylor Series Approximation of order $n$ to the ($n$-times differentiable) function $f\left(x\right)$ is given by:

$f_n\left(x\right) \= f\left(a\right) \+f\'\left(a\right)\left(x-a\right)\+\frac{\mathrm{f"}\left(a\right)}{2\!}{\left(x-a\right)}^2\+ ..\. \+\frac{f^{\left(n\right)}\left(a\right)}{n\!}{\left(x-a\right)}^n$. 

Taylor's Theorem essentially states that $f_n$ is the best possible degree $n$ polynomial approximation to $f$ about the point $a$. Specifically, the difference, $f\left(x\right)-f_n\left(x\right)$ can be written in the form:

$f\left(x\right)-f_n\left(x\right)\=h_n\left(x\right)\cdot {\left(x-a\right)}^n$, where $\underset{x\to a}{lim}{h}_n\left(x\right)\=0$.

The special case of Taylor Series in which $a \= 0$ is known as the Maclaurin Series.

$f{\left(x\right)}_{ }\=$sin(x)cos(x)x^3 sinh(x)cosh(x)

$a \=$

$\mathrm{order} n{ }_{ }\=$

$f_n\left(x\right) \=$

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