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April 16th, 2012, 06:26 PM   #1
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Proof of divergence for the harmonic series

Book says that step one is to show that for n > 1



I know how to algebraic show that this is the same as

And this is the same as (1) Now the book shows in the solution that the next step is (2)

How did they go from (1) to (2) algebraically? I thought it would be by using the method of partial fractions but when I did that I obtained which is not correct.
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April 16th, 2012, 06:33 PM   #2
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Re: Proof of divergence for the harmonic series

You have made an error in your partial fraction decomposition. You could get to (2) by beginning with:



Multiply through by n



(2)
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April 16th, 2012, 06:38 PM   #3
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Re: Proof of divergence for the harmonic series

That makes perfect sense but why in the heck did they bother mentioning the first step to be ? Then they went to (2). I just don't understand why they did it this way instead of your method. Do you know why?
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April 16th, 2012, 06:41 PM   #4
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Re: Proof of divergence for the harmonic series

No, I couldn't say, but maybe they are going to use (1) at some point later in the proof. I doubt that was done unnecessarily.
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April 16th, 2012, 06:47 PM   #5
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Re: Proof of divergence for the harmonic series

So does what you have shown prove that for n > 1

?

If so, why does it prove it? I don't understand how solving for n proves the statement above.
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April 16th, 2012, 06:49 PM   #6
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Re: Proof of divergence for the harmonic series

Should be

and I did not mean solve for n. I meant having n on one side of the equation.
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April 16th, 2012, 07:00 PM   #7
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Re: Proof of divergence for the harmonic series

If I were going to prove the original statement, I would write it in the form:





We know this is true for any finite value of 1 < n.
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April 17th, 2012, 11:02 AM   #8
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Re: Proof of divergence for the harmonic series

Ok then next part of the question says: By grouping the terms of the series by threes, verify the inequality

1 + (1/2 + 1/3 + 1/4) + (1/5 + 1/6 + 1/7) + ... > 1 + 3/3 + (3/6 + 3/9 + 3/12) + ... > 1 + 3/3 + 9/9 + 27/27 + ...

How do I even begin to verify the inequality? Also, I am confused on what exactly it is asking me to verify. I am guessing it wants me to verify that it holds for any n using induction?
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April 17th, 2012, 11:46 AM   #9
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Re: Proof of divergence for the harmonic series

We have verified:



Hence we know:







etc.

Now, since we also have:



which is required for the last step in the given inequality.
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April 18th, 2012, 10:33 AM   #10
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Re: Proof of divergence for the harmonic series

Quote:
Originally Posted by MarkFL
If I were going to prove the original statement, I would write it in the form:



I did you go from our original statement to this? I thought I understood it yesterday but I cannot seem to manipulate our given information into this.
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