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Prove that u_n \to \infty [IMO Longlist 1983]

Source: IMO Longlist 1983

October 7, 2010
functionlimitlogarithmsalgebra unsolvedalgebra

Problem Statement

Let ff be a real-valued function defined on I=(0,+∞)I = (0,+\infty) and having no zeros on II. Suppose that lim⁡x→+∞f′(x)f(x)=+∞.\lim_{x \to +\infty} \frac{f'(x)}{f(x)}=+\infty. For the sequence un=ln⁡∣f(n+1)f(n)∣u_n = \ln \left| \frac{f(n+1)}{f(n)} \right|, prove that un→+∞u_n \to +\infty as n→+∞.n \to +\infty.
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