I cannot accept that $\sum_{n=1}^\infty n = -\frac{1}{12}$. It should be that such a sum is divergent. That it is divergent is useful for the Test for Divergence in many such problems. I feel like the sum of -1/12 does something wrong. What is actually happening?
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3Possibly already answered at http://math.stackexchange.com/questions/39802/why-does-123-cdots-frac112 – ForgotALot Apr 15 '16 at 05:58
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With the usual definition of convergence of a series (that is, convergence of partial sums), then this seris is not convergent. There is absolutely no doubt about this. However, there are "transformations of series" (such as Cesàro or Borel summation) that can turn some divergent series into convergent ones. See also Divergent series on Wikipedia. – Jean-Claude Arbaut Apr 15 '16 at 06:18
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@Jean-ClaudeArbaut this helps, thanks! – change_picture Apr 15 '16 at 06:23
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What is actually happening is that there are ways to assign a real number to series in such a way that when the series happens to converge, that number coincides with the sum. As such, we call them "sums" even though they are some times not. Abel summation and Cesaro summation are two examples, as well as analytic continuation of the Riemann-zeta function or even hand-wavey term-by-term manipulation. Most such methods that work for this specific series give the result $-\frac1{12}$.
Also note that this has been "experimentally verified" in the sense that when using quantum mechanics to calculate the strength of the physical phenomenon known as the Casimir effect, you come across this sum. If you declare the sum to be $-\frac1{12}$, you get a final answer that agrees with what is measured in a laboratory.
Arthur
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But the series simply cannot converge. I'm okay with Riemann's proof that for any conditionally convergent series, and for any real number r, there is a rearrangement of the terms in the series that makes makes the sum r. However, we're not even working with something that is conditionally convergent...except apparently we are? I'm so confused. I guess I need to read more about this strange stuff. – change_picture Apr 15 '16 at 06:07
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@gorzardfu The series does not converge. This is clear. However, as I said, there are ways of assigning some real number to a given series other than just taking its sum. Some such methods agree with the conventional sum for all converging series, but gives a finite result for some non-converging ones. That is what is happening here, and it's not all that strange. – Arthur Apr 15 '16 at 06:13
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