(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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].

2. Relevant equations

3. 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 isnothomework), 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?

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# Second order Taylor expansion

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