Taylor series

For a function, $LaTeX: f,$ having all derivatives, the series is:

$LaTeX: \sum_{k=0}^{\infty} \frac{f^{(k)} (x + h)}{k!}\, h^k,$

where $LaTeX: \textstyle f^{(k)}$ is the k-th derivative of $LaTeX: f.$ Truncating the series at the n-th term, the error is given by:

$LaTeX: \varepsilon_n (h) = \left \vert f(x) - \sum_{k=0}^{n} \frac{f^{(k)} (x + h)}{k!}\, h^k \right \vert.$

This is a Taylor expansion, and for the Taylor series to equal the functional value, it is necessary that the error term approaches zero for each n:

$LaTeX: \lim_{h \to 0} \varepsilon_n (h) = 0.$

In that case, there must exist $LaTeX: y$ in the line segment $LaTeX: \textstyle [x,x+h]$ such that

$LaTeX: \varepsilon_n (h) = \frac{f^{(n+1)} (y)}{(n + 1)!} h^{n+1} .$