(i) Prove Taylor's Theorem for a function $f: \mathbb{R} \rightarrow \mathbb{R}$ differentiable $n$ times, in the following form: for every $x \in \mathbb{R}$ there exists $\theta$ with $0<\theta<1$ such that

$$f(x)=\sum_{k=0}^{n-1} \frac{f^{(k)}(0)}{k !} x^{k}+\frac{f^{(n)}(\theta x)}{n !} x^{n}$$

[You may assume Rolle's Theorem and the Mean Value Theorem; other results should be proved.]

(ii) The function $f: \mathbb{R} \rightarrow \mathbb{R}$ is twice differentiable, and satisfies the differential equation $f^{\prime \prime}-f=0$ with $f(0)=A, f^{\prime}(0)=B$. Show that $f$ is infinitely differentiable. Write down its Taylor series at the origin, and prove that it converges to $f$ at every point. Hence or otherwise show that for any $a, h \in \mathbb{R}$, the series

$$\sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k !} h^{k}$$

converges to $f(a+h)$.