Paper 1, Section II, F

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be $n$-times differentiable, for some $n>0$.

(a) State and prove Taylor's theorem for $f$, with the Lagrange form of the remainder. [You may assume Rolle's theorem.]

(b) Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}$ is an infinitely differentiable function such that $f(0)=1$ and $f^{\prime}(0)=0$, and satisfying the differential equation $f^{\prime \prime}(x)=-f(x)$. Prove carefully that

$f(x)=\sum_{k=0}^{\infty}(-1)^{k} \frac{x^{2 k}}{(2 k) !}$

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