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August 4th, 2012, 04:47 AM   #1
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A challenge to the MMF members

Prove that

using calculus.

Balarka

.
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August 4th, 2012, 05:04 AM   #2
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Re: A challenge to the MMF members

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August 4th, 2012, 05:54 AM   #3
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Re: A challenge to the MMF members

Quote:
Originally Posted by ZardoZ
Actually, yes you are right.

But the Dirichlet Series for and can be regularized by this calculation:





implies

And so,



This is the algebraic calculation.
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August 4th, 2012, 05:58 AM   #4
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Re: A challenge to the MMF members

If you are interested, I could post my method which uses differential equations involving trigonometric functions.
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August 4th, 2012, 06:29 AM   #5
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Re: A challenge to the MMF members

This is actually the famous Grandi's series.

My calculations:



Satisfies the differential equation:

.

Now put f(x) = sin(x). A well-known solution to is

And by (1), a particular solution is

So,

Now put x = 0. eliminate all sin(0) = 0. substitute cos(0) = 1:



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August 4th, 2012, 06:43 AM   #6
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Re: A challenge to the MMF members

Another way to handle the Grandi's series is to use the relation:



put z = 0,

by the analytic continuation of the zeta function.
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August 4th, 2012, 01:12 PM   #7
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Re: A challenge to the MMF members

The article says you can rearrange the terms to arrive at any integer solution. I wonder if you can rearrange the terms to arrive at ?

Using this order... the sum alternates between 0 and 1

now you regroup like this..

again the sum INSIDE parentheses alternates between 0 and 1, so this second series with different grouping will be



or



the 1/2 you got is the AVERAGE of 0 and 1 because you used a variable S to represent an indeterminate value, but variables are supposed to represent unknown determinate (convergent) values, then we can be sure that legal algebraic operations may lead to a correct result.

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August 4th, 2012, 09:12 PM   #8
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Re: A challenge to the MMF members

Quote:
Originally Posted by agentredlum
The article says you can re-arrange the terms to arrive at any integer solution, i wonder if you can re-arrange the terms to arrive at ?

Using this order... the sum alternates between 0 and 1

now you regroup like this..

again the sum INSIDE parenthesis alternates between 0 and 1 so this second series with different grouping will be



or



the 1/2 you got is the AVERAGE of 0 and 1 because you used a variable S to represent an indeterminate value, but variables are supposed to represent unknown determinate (convergent) values, then we can be sure that legal algebraic operations may lead to a correct result.

I know that Grandi's formulation is rather questionable.
But what about the analytic continuation of the zeta function? It clearly says that
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August 5th, 2012, 12:35 AM   #9
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Re: A challenge to the MMF members

Well, I don't understand too much about analytic continuation of the zeta function. i don't trust the result

By definition...

now if you put in s = 0 you should get

which is clearly 1+1+1+ ... -> divergent

how do they get -1/2 out of this?

they must be using another definition of zeta that i do not understand...

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August 5th, 2012, 12:40 AM   #10
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Re: A challenge to the MMF members

Quote:
Originally Posted by agentredlum
they must be using another definition of zeta that i do not understand...
[color=#000000]I have seen such strange proofs of such results in Ramanujan's notebooks, but to tell you the truth my knowledge of mathematics is not so advanced. I like to keep my mathematics to Earthly grounds. [/color]
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