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November 10th, 2013, 10:03 AM   #1
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1st year math - Riemann zeta function & series

the riemann zeta function is defined for p > 1 as

prove by considering

so my thoughts: as then we just use the definition of convergence to prove that the partial sum converges to 1 and we're done

however I'm having troubles trying to show that for some epsilon there exists a real number H s.t. P>H and N>H implies

I've tried using geometric series for the inside sum and then trying to rearrange for P or N but to no avail - any help please
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November 10th, 2013, 12:42 PM   #2
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Re: 1st year math - Riemann zeta function & series

Have you noticed that:

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November 10th, 2013, 01:28 PM   #3
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Re: 1st year math - Riemann zeta function & series

Quote:
Originally Posted by Pero
Have you noticed that:

I have not reached a stage where I can use that - could you please show me where?
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November 10th, 2013, 01:42 PM   #4
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Re: 1st year math - Riemann zeta function & series

If you re-arrange into geometric series:



Then, each sums to:



Does that help?
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November 10th, 2013, 01:58 PM   #5
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Re: 1st year math - Riemann zeta function & series

Quote:
Originally Posted by Pero
If you re-arrange into geometric series:



Then, each sums to:



Does that help?
I understand what you're doing but I'm considering the partial sums, why are you taking P to infinity?
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November 10th, 2013, 02:20 PM   #6
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Re: 1st year math - Riemann zeta function & series

Quote:
Originally Posted by lun123
[
why are you taking P to infinity?
Where else would you like to take it? N's got to get there too!
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November 11th, 2013, 11:21 AM   #7
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Re: 1st year math - Riemann zeta function & series

I'm sorry but I think you misunderstood what I was asking:

I'm asking, how do I show that for all there exists a real number H s.t. P>H and N>H implies
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