Analysis I | Part IA, 2002

Prove Taylor's theorem with some form of remainder.

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


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

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

for all xx with xR|x| \leqslant R. Explain why we can find an MM such that

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

for all xx with xR|x| \leqslant R and all j0j \geqslant 0.

Use your form of Taylor's theorem to show that

f(x)=n=0x3n(3n)!f(x)=\sum_{n=0}^{\infty} \frac{x^{3 n}}{(3 n) !}

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