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I recently learned about various techniques for summing divergent series. These techniques make it possible to show that, for example, $$1 + 2 + 3 + 4 + 5 + ... = \frac{-1}{12}.$$

I am a bit confused because the summation $1 + 2 + 3 + 4 + 5 + ...$ diverges under the definition of infinite summations I'm used to (namely, as a limit of the finite summations) but still seems to work out to a number using these other techniques. I'm trying to determine how to reconcile these disparate pieces of information together.

Right now, I have two different hypotheses about how to understand these results together simultaneously:

  1. There are two different definitions of what an infinite summation "means." One uses limits of finite summations, and the other uses complex analysis. Since these definitions aren't identical to one another, it's not surprising that they predict different results in this case, and the disparity arises because there are two different definitions that happen to use similar notation to describe their results.

  2. Just because the partial sums of the first 0, 1, 2, 3, 4, 5, ... etc. terms of the series don't converge to a value doesn't mean that the sum of all infinitely many terms doesn't have a value. The series diverges because the finite partial sums don't converge to anything, but the infinite summation really is indeed $\frac{-1}{12}$.

Are either of these hypotheses correct? Or am I off-base here? I'm hoping to learn how to think about results like these, and if there's some bigger picture that everything fits into I'd appreciate more information about it.

Thanks so much!

WLOG
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templatetypedef
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    This is almost identical to the question you asked 2 mins ago. – Lost1 Jan 12 '14 at 02:49
  • @Lost1- I asked these questions separately because I was under the impression that they were getting at two separate concepts. My earlier question was about whether a specific proof involving 1 + 1 - 1 + 1 ... contained a specific flaw that I had pointed out. This question was (supposed to be) about a more general understanding of how to reconcile the fact that divergent series can sum to a limit. I'm really new to this, so I apologize if the answers are the same in both cases. – templatetypedef Jan 12 '14 at 02:51
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    Look up http://en.wikipedia.org/wiki/Divergent_series – Robert Israel Jan 12 '14 at 02:54
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    This has been asked before many times. Your second bulletpoint is not good, though. "The sum of infinitely many values" cannot "converge" to something, a sequence of partial sums does. – Pedro Jan 12 '14 at 02:56
  • @PedroTamaroff- Thanks for pointing out the issue in the second bullet point. I didn't find any questions like this one before asking it, and I'm sorry if it's a duplicate. I saw some other questions asking why the summation came out to $\frac{-1}{12}$, and I thought this one was different since it was more along the lines of "how do you interpret the result?" versus "what math do you use to show that the sum comes out this way." If there's another question like this, do you know how I could find it? – templatetypedef Jan 12 '14 at 03:01
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    At this point, you may be able to find other questions like this by looking at the list on this page, under the heading Related. – Gerry Myerson Jan 12 '14 at 03:51
  • Hypothesis 1 is correct, There are actually many ways of defining "the sum" of an infinite series, and as @RobertIsrael linked, the wikipedia article for Divergent series can show you some of them. – Mark S. Jan 12 '14 at 16:46
  • This question spawned a discussion on meta – Grigory M Jan 12 '14 at 21:13
  • The only definition is that the limit of the partial sums exists. You can come come to all sort of rediculous conclusions it you assume that the limit exists when it actually doesn't. So, saying "Let $n = 1+2+3\cdots$...$ when n doesn't exists can lead to all sorts of conclusions that may be true and may be false. – Steven Alexis Gregory Mar 09 '16 at 16:36
  • You get this result by analytic continuation of the Riemann zeta function. Watch this video to see it graphically: https://www.youtube.com/watch?v=sD0NjbwqlYw – Stefan Gruenwald Mar 19 '17 at 03:37

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