December 8th, 2010, 12:01 PM  #1 
Senior Member Joined: Apr 2009 Posts: 106 Thanks: 0  Harmonic Series
1/a + 1/(a+1) + 1/(a+2) + ... + 1/(a^2) > 1 Why does this prove that the harmonic series diverges to infinity? What is so important about the finite number of terms? 
December 8th, 2010, 12:14 PM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Harmonic Series
1 + 1/2 + 1/3 + 1/4 + ... = (1) + (1/2 + 1/3 + 1/4) + (1/5 + ...) + ... >= 1 + 1 + 1 + ...

December 8th, 2010, 12:38 PM  #3 
Senior Member Joined: Nov 2010 Posts: 502 Thanks: 0  Re: Harmonic Series
The idea is that if you have an infinite number of finite sets, each of which add to a single number, then the total sum must diverge (assuming, of course, that the terms are all positive). Alternately, the harmonic series diverges and so it is not bounded by any number. Let's say we know that adding up the first k elements of the series gives us some number above the integer N. Then I know that adding the next (k+1)^2 elements will be above the integer N+1. In this fashion, one can see the series gets arbitrarily large.


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