Paper 1, Section II, F

Analysis I | Part IA, 2010

(a) State and prove Taylor's theorem with the remainder in Lagrange's form.

(b) Suppose that e:RRe: \mathbb{R} \rightarrow \mathbb{R} is a differentiable function such that e(0)=1e(0)=1 and e(x)=e(x)e^{\prime}(x)=e(x) for all xRx \in \mathbb{R}. Use the result of (a) to prove that

e(x)=n0xnn! for all xRe(x)=\sum_{n \geqslant 0} \frac{x^{n}}{n !} \quad \text { for all } \quad x \in \mathbb{R}

[No property of the exponential function may be assumed.]

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