User:Matt Kwan/Cauchy

Lagrange form is proved here: https://docs.google.com/viewer?url=http://www.math.hawaii.edu/~jb/math216/lagrange.pdf&pli=1

To prove: $R_{n}(x)={\frac {f^{(n+1)}(\xi )}{n!}}x(x-\xi )^{n}$ where $f$ is some $n+1$ times differentiable function, $p_{n}$ is the $n$th order taylor polynomial for $f$, and $\xi \in (0,x)$

set $x$, set $n$

${\begin{aligned}&g(t)=\sum _{k=0}^{n}{\frac {f^{(k)}(t)}{k!}}(x-t)^{k}\\&g(x)=f(x)\\&g(0)=p_{n}(x)\\\end{aligned}}$

By the mean value theorem:

${\begin{aligned}g'(\xi )&={\frac {g(x)-g(0)}{x-0}}\\&={\frac {R_{n}(x)}{x}}\end{aligned}}$

By the product rule:

${\begin{aligned}g'(t)&=f(t)+\sum _{k=1}^{n}\left({\frac {f^{(k+1)}(t)}{k!}}(x-t)^{k}-{\frac {f^{(k)}(t)}{(k-1)!}}(x-t)^{k-1}\right)\\&=\sum _{k=0}^{n}{\frac {f^{(k+1)}(t)}{k!}}(x-t)^{k}-\sum _{q=0}^{n-1}{\frac {f^{(q+1)}(t)}{q!}}(x-t)^{q}\\&={\frac {f^{(n+1)}(t)}{n!}}(x-t)^{n}\\\end{aligned}}$

${\begin{aligned}R_{n}(x)={\frac {f^{(n+1)}(\xi )}{n!}}x(x-\xi )^{n}\\\end{aligned}}$