# Taylor's Theorem

Taylor's theorem shows that a function can be approximated within some region to within some error by a polynomial. Specifically

Note that the function must be sufficiently differentiable, meaning that all of the derivatives of \(f\) up to the \((n+1)^{th}\) must exist.

\[
f(x+\delta) = f(x) + \delta \, f'(x) + \frac 1 2 \delta^2 \, f''(x) + ... + \frac 1 {n!} \delta^n \, f^{(n)}(x) + O\left(\delta^{n+1}\right)
\]

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.