Exercise 2, § 4

General Topology, V

Show, by induction on $n$ , that for all integers $n > 0$ we have

2^n > \frac{1}{2} n(n+1).

Deduce that, if

a > 1

and

\alpha > 0

\lim_{x \to +\infty} \frac{a^x}{x^\alpha} = +\infty, \quad \lim_{x \to +\infty} \frac{\log_a x}{x^\alpha} = 0

(begin by proving the first of these relations for

a = 2

and

\alpha = 1

Solution

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