# Thread: integrals and taylor's remainder function

1. ## integrals and taylor's remainder function

I am stumped as to how to approach this one:
Prove that the remainder term in Taylor's theorem can be replaced by
$R_{n}(x)=\frac{1}{n!}\int^x_a f^{(n+1)}(t)(x-t)^ndt$ provided that $f^{(n+1)}$ is Riemann integrable.

[Taylors remainder term: $R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$

2. Use integration by parts with $u'=f^{(n+1)}(t)$ and $v=(x-t)^n$ and when done rememeber that $R_{n-1,a}(f)-R_{n,a}(f)=P_{n,a}(f)-P_{n-1,a}(f)= \frac{ f^{(n)}(a)(x-a)^n}{n!}$

PS. Notice that behind all of this the argument is inductive so you should try to prove the case n=1,2 apart