Paper 1, Section II, F

Analysis I | Part IA, 2021

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

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

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

f(x)=k=0(1)kx2k(2k)!f(x)=\sum_{k=0}^{\infty}(-1)^{k} \frac{x^{2 k}}{(2 k) !}

Typos? Please submit corrections to this page on GitHub.