The Meaning of Divergent Sums

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Heptadecagon
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The Meaning of Divergent Sums

Postby Heptadecagon » Mon May 28, 2012 7:20 pm UTC

I have in the past often heard about divergent sums such as 1+2+3+4... = -1/12 , but became very interested when I discovered my own proof using relatively simple methods. Is there any way to get a explanation/intuition about these things, or reason why so many different methods evaluate these sums the same? I'm curious as to how this works, all I can think of is the number line 'looping around at infinity', although that idle thought isn't satisfying. And are such statements actually true in every sense, or just as sort of... consistent sense? I have many questions, thanks for any insight you may provide.

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z4lis
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Re: The Meaning of Divergent Sums

Postby z4lis » Tue May 29, 2012 6:33 pm UTC

If I remember correctly, results like that are found using analytic continuation of functions. For instance, we might have a function like 1^s + 2^s + 3^s + ... that converges for Re(s) < 1 that we can extend through the complex plane, analytically. Now analytic continuation is unique. Once the function is specified on one region, there's only one way to extend it analytically, so if you try and do it two different ways, you'll always get the same answer.The function above can be analytically extended to the point s=1, where presumably it has the value -1/12. Now the original definition of the function, 1^s + 2^s + 3^s + ... doesn't work at s = 1. But there's a continuation that does, so we can abuse notation and write something like 1 + 2 + 3 + ... = -1/12.

http://en.wikipedia.org/wiki/Riemann_zeta_function might interest you, as well as http://en.wikipedia.org/wiki/Analytic_continuation
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.


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