Taylor’s theorem

1 Taylor’s Theorem

Let $f$ be a function which is defined on the interval $(a,b)$ and suppose the $n$th derivative $f^{(n)}$ exists on $(a,b)$. Then for all $x$ and $x_0$ in $(a,b)$,

$$R_n(x)=\frac{f^{(n)}(y)}{n!}{(x-x_0)}^n$$

with $y$ strictly between $x$ and $x_0$ ($y$ depends on the choice of $x$). $R_n(x)$ is the $n$th remainder of the Taylor series for $f(x)$.