Define a_{n} = 2(1 + 1/n)^{2n+1}/( (1 + 1/n)^{n} + (1 + 1/n)^{n+1}). Prove that a_{n} is strictly monotonic increasing.

**Solution**

a_{n} = 2 (n+1)^{n+1}/(n^{n} (2n+1) ). Let f(n) = ln a_{n}/2 = (n + 1) ln(n + 1) - n ln n - ln(2n+1). Regard f as a function of a real variable and differentiate. f '(x) = ln(x + 1) - ln x - 2/(2x + 1). Differentiate again: f ''(x) = 1/(x + 1) - 1/x + 4/(2x + 1)^{2} = -1/( x(x+1)(2x+1)^{2}).

So f ''(x) < 0 for all x > 0. Hence f '(x) is decreasing. But we can write f '(x) = ln(1 + 1/x) - 2/(2x + 1) which tends to 0 as x tends to infinity. So f '(x) > 0 for all x > 0. Hence f(x) is strictly increasing for all positive x. Hence f(n+1) > f(n) and so exp f(n+1) > exp f(n), which is the result we want.

© John Scholes

jscholes@kalva.demon.co.uk

5 Mar 2002