Let http://www.cramster.com/Answer-Board...6185762832.gif with n>1. Prove that 1 + 1/2 + 1/3 + .....1/n http://www.cramster.com/Answer-Board...6025747153.gif

Printable View

- Sep 13th 2008, 09:53 AMrmpatel5number theory proof
Let http://www.cramster.com/Answer-Board...6185762832.gif with n>1. Prove that 1 + 1/2 + 1/3 + .....1/n http://www.cramster.com/Answer-Board...6025747153.gif

- Sep 13th 2008, 10:00 AMrmpatel5number theory proof
Let http://www.cramster.com/Answer-Board...6185762832.gif with n>1. Prove that 1 + 1/2 + 1/3 + .....1/n http://www.cramster.com/Answer-Board...6025747153.gif

- Sep 13th 2008, 11:52 AMPaulRS
Let be the greatest prime not exceeding .

So we must have: (1) . Suppose this were false, then but, by Bertrand's Postulate there's a prime such that which is absurd since is the greatest prime not exceeding .

Now suppose then since is an integer

By (1) we have that doesn't divide so is not an integer. Absurd - Sep 13th 2008, 01:10 PMAryth
There is a classical proof provided by Nicole Oresme, where you consider the subsequence :

And therefore, in general:

.

And since the subsequence is unbounded, the sequence diverges.

And therefore it is sufficient to say that if the sequence diverges, no integer can be reached and therefore:

- Sep 13th 2008, 02:10 PMawkward
- Sep 13th 2008, 02:19 PMo_O

The sequence diverges but you can't conclude S isn't an integer for all . - Sep 13th 2008, 05:17 PMThePerfectHacker
It is also here.