Prove Taylor's theorem with some form of remainder.

An infinitely differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfies the differential equation

$$f^{(3)}(x)=f(x)$$

and the conditions $f(0)=1, f^{\prime}(0)=f^{\prime \prime}(0)=0$. If $R>0$ and $j$ is a positive integer, explain why we can find an $M_{j}$ such that

$$\left|f^{(j)}(x)\right| \leqslant M_{j}$$

for all $x$ with $|x| \leqslant R$. Explain why we can find an $M$ such that

$$\left|f^{(j)}(x)\right| \leqslant M$$

for all $x$ with $|x| \leqslant R$ and all $j \geqslant 0$.

Use your form of Taylor's theorem to show that

$$f(x)=\sum_{n=0}^{\infty} \frac{x^{3 n}}{(3 n) !}$$