42nd Eötvös 1938

Show that 1/n + 1/(n+1) + 1/(n+2) + ... + 1/n² > 1 for integers n > 1.

Solution

There are n²-n terms after the first. These terms are all at least 1/n² and at least one of them is > 1/n², so their sum > (n²-n)/n² = 1 - 1/n. Hence the sum of all the terms > 1.

42nd Eötvös 1938

© John Scholes
jscholes@kalva.demon.co.uk
1 Nov 2003
Last corrected/updated 1 Nov 03