Prove that 12n leq 1352n1242n whenever n is a positive integ

Prove that    \\(1/(2n) \\leq [1*3*5*****(2n-1)]/(2*4*****2n)\\) whenever n is a positive integer.

Solution

For convenience in the following working, let

f(n) = 1/(2n) and

g(n) = [1*3*5*...*(2n-1)]/(2*4*...*2n).

For the base step: f(1) = 1/2 = g(1).

For the inductive step, we must show that if f(n) ? g(n), then f(n+1) ? g(n+1).

One way to do this is to calculate f(n+1)/f(n) and g(n+1)/g(n).

Since all the terms we\'re dealing with are positive numbers, if we can show that, for n ? 1, f(n+1)/f(n) ? g(n+1)/g(n), the inductive step will follow. (Convince yourself why this is true.)

f(n+1)/f(n) = [1/(2n+2)] / [1/(2n)] = 2n/(2n+2) = n/(n+1).

g(n+1)/g(n) = [1*3*5*...*(2n-1)*2n]/(2*4*...*2n*(2n+2)... / [[1*3*5*...*(2n-1)]/(2*4*...*2n)]

= 2n/(2n+2) = n/(n+1).

Thus, for n ? 1, f(n+1)/f(n) ? g(n+1)/g(n), and hence f(n) ? g(n) implies f(n+1) ? g(n+1), and the result follows by mathematical induction.

Notice that we found f(1) = g(1) and f(n+1)/f(n) = g(n+1)/g(n), so in fact we have proved that 1/(2n) = [1*3*5*...*(2n-1)]/(2*4*...*2n), a stronger result than inequality!