In Theorem 1.15 prove that (E_{p}(x, delta)=oleft(delta^{p}ight)), as (delta ightarrow 0). Theorem 1.15. Let f be a

In Theorem 1.15 prove that \(E_{p}(x, \delta)=o\left(\delta^{p}ight)\), as \(\delta ightarrow 0\).

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Theorem 1.15. Let f be a function that has p+ 1 bounded and continuous derivatives in the interval (x, x+8). Then f(x+8)= P Sk f(k) (x) + Ep(x, 8), k! k=0 where Ep(x, 8) = O(SP+1) and Ep(x, 8) = o(SP) as 8 0. Proof. We will prove that Ep(x, 8) = O(SP+1) as 80. The fact that Ep(x, 8)o(8P) is proven in Exercise 34. From Theorem 1.13 we have that = SP+1 f(p+1)() Ep(x, 8) = (p+1)!