- #1
AxiomOfChoice
- 533
- 1
Homework Statement
Show that if [itex]F[/itex] is twice continuously differentiable on [itex](a,b)[/itex], then one can write
[tex]
F(x+h) = F(x) + h F'(x) + \frac{h^2}{2} F''(x) + h^2 \varphi(h),
[/tex]
where [itex]\varphi(h) \to 0[/itex] as [itex]h\to 0[/itex].
Homework Equations
The Attempt at a Solution
I'm posting this here because it's a problem in Stein-Shakarchi's "Fourier Analysis". I'm working through this book on my own (so this problem is not homework), but I thought it'd look suspicious if I posted it in the regular forums.
I believe I've managed to show that
[tex]
F(x+h) = F(x) + h F'(x) + \frac{h^2}{2} F''(x) + \int_0^h w \psi(w) dw,
[/tex]
where
[tex]
\psi(h) = \frac{F'(x+h) - F'(x)}{h} - F''(x),
[/tex]
but I'm not sure how I'm supposed to go about showing that
[tex]
\int_0^h w \psi(w) dw = h^2 \varphi(h).
[/tex]
What do you think the [itex]\varphi(h)[/itex] they're wanting here is?