I find the idea behind this theorem is somewhat difficult to grasp.(adsbygoogle = window.adsbygoogle || []).push({});

Anyhow, there is a related problem in Rudin that I can't figure out.

So let f be continuously differentiable n-1 times on [a,b] where the nth derivative exists on (a,b). z and a are two distinct points of the interval. Put

Q(t)=(f(t)-f(z))/(t-z)

Let P(t) be the (n-1)th degree taylor polynomial of f about a. Show that

f(z)=P(z) + [Q^(n-1)(a)(z-a)^n]/(n-1)!

I've tried expanding this every way I can think of but am not getting the result. I can't even see why it's true.

My 2nd question regards analyticity vs. C^infinity. So e^(-1/x) and 0 at 0 shows these aren't the same concepts. What does it mean to say that r(h) (the remainder function) goes to zero faster than h does ie what does r(h)/h goes to zero as h goes to zero show/mean?

Taylor's theorem was not emphasised in my singe variable analysis class and we took it as given in my complex analysis class - so it's relatively new to me and I don't have an intuitive understanding of the concept.

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# Taylor's Theorem

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