Hello, The harmonic numbers H_{k}, k = 1,2,3.....are defined by H_{k} = 1 + 1/2 + 1/3....1/k I am trying to prove by mathematical induction: H_{2n} >= 1 + n/2 , whenever n is a nonnegative integer. H_{8} = H_{23} >= 1 + 3/2 Can you help? Thank you in advance!!!! :) Hi Leslie,In the inductive step you assume that H_{2n} >= 1 + n/2 and you need to show that H_{2n+1} >= 1 + (n+1)/2. The proof is as follows: = 1 + 1/2 + 1/3+...+1/2^{n} + 1/(2^{n} + 1) + 1/(2^{n} + 2)+...+ 1/2^{n+1} = H_{2n} + 1/(2^{n} + 1) + 1/(2^{n} + 2)+...+ 1/2^{n+1} >= 1 + n/2 + 1/(2^{n} + 1) + 1/(2^{n} + 2)+...+ 1/2^{n+1} The latter step follows by the inductive assumption. In the line above each of the denominators, 2^{n} + 1, 2^{n} + 2, ... is less than 2^{n+1} and hence each of the fractions 1/(2^{n} + 1), 1/(2^{n} + 2),... is greater than 1/2^{n+1}. Thus H_{2n+1} >= 1 + n/2 + 1/(2^{n} + 1) + 1/(2^{n} + 2)+...+ 1/2^{n+1} >= 1 + n/2 + 1/2^{n+1} + 1/2^{n+1}+...+ 1/2^{n+1} = 1 + n/2 + 2^{n}/2^{n+1} = 1 + n/2 + 1/2 = 1 + (n+1)/2 Harley
