# 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 $f(x)$.

Title Taylor’s theorem TaylorsTheorem 2013-03-22 11:56:53 2013-03-22 11:56:53 Andrea Ambrosio (7332) Andrea Ambrosio (7332) 11 Andrea Ambrosio (7332) Theorem msc 41A58 TaylorSeries