# Taylor's Theorem

Taylor's theorem shows that a function can be approximated within some region to within some error by a polynomial. Specifically
$f(x+\delta) = f(x) + \delta \, f'(x) + \frac 1 2 \delta^2 \, f''(x) + ... + \frac 1 {n!} \delta^n \, f^{(n)}(x) + O\left(\delta^{n+1}\right)$
where $$f'(x)$$ is the derivative of $$f$$ evaluated at $$x$$, $$f''(x)$$ the second derivative and $$f^{(n)}(x)$$ the $$n^{th}$$. The exclamation mark is the factorial of the value preceding it and $$O(\delta^{n+1})$$ is an error term of order $$\delta^{n+1}$$ or, in other words, is for any given $$f$$ and $$x$$ is equal to some constant multiple of $$\delta^{n+1}$$.
Note that the function must be sufficiently differentiable, meaning that all of the derivatives of $$f$$ up to the $$(n+1)^{th}$$ must exist.

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