Paper 1, Section I, D

Analysis I | Part IA, 2013

Show that exp(x)1+x\exp (x) \geqslant 1+x for x0x \geqslant 0.

Let (aj)\left(a_{j}\right) be a sequence of positive real numbers. Show that for every nn,

1naj1n(1+aj)exp(1naj)\sum_{1}^{n} a_{j} \leqslant \prod_{1}^{n}\left(1+a_{j}\right) \leqslant \exp \left(\sum_{1}^{n} a_{j}\right)

Deduce that 1n(1+aj)\prod_{1}^{n}\left(1+a_{j}\right) tends to a limit as nn \rightarrow \infty if and only if 1naj\sum_{1}^{n} a_{j} does.

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