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December 8th, 2010, 12:01 PM   #1
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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?
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December 8th, 2010, 12:14 PM   #2
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Re: Harmonic Series

1 + 1/2 + 1/3 + 1/4 + ... = (1) + (1/2 + 1/3 + 1/4) + (1/5 + ...) + ... >= 1 + 1 + 1 + ...
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December 8th, 2010, 12:38 PM   #3
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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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