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In this youtube video the speakers present a line of reasoning as to why $$ \sum_{n=1}^\infty n = -\frac{1}{12} $$

in other words

$$1+2+3+\cdots=-\frac{1}{12}$$ and my brain still hurts. It doesn't make any sense but it was also shown by other mathematicians like Euler.

They said that this answer shows itself in quantum physics in explaining the Casimir effect and it has a role in explaining why there are 26 dimensions in string theory.

But, I don't think we need to delve into quantum physics and string theory to find examples of phenomena where we get negative results for positive infinite sums. The practically infinite sum of money, time, and effort to put a child through school and college, then the negative result of said child unable to obtain gainful employment. Economics immediately came to mind. I did a google search on this but nothing relevant came up.

Can the Ramanujan sum be used to explain unexpected outcomes in macro-economics? For example: return on investment for government spending

student debt for college education

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    $\begingroup$ You are way out of context here. This famous sum you talk about is intimately related to complex numbers and the Riemann zeta function. These topics are usually dealt with in complex analysis and bear hardly any economic relevance. Also, your question has nothing to do with government-debt or government-spending, so please don't include those tags. $\endgroup$
    – Taufi
    Jun 2, 2017 at 13:48
  • $\begingroup$ If it is true and proven to be true then it must be relevant in all contexts. $\endgroup$ Jun 2, 2017 at 14:00
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    $\begingroup$ I'm voting to close this question as off-topic because it's not about economics. It's nonsense based on a misunderstanding $\endgroup$
    – 410 gone
    Jun 2, 2017 at 15:06
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    $\begingroup$ I'm voting to close this question as off-topic because it does not seem to have anything to do with economics beyond asking "does this have anything do with economics?" $\endgroup$
    – Giskard
    Jun 2, 2017 at 15:06
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    $\begingroup$ That line of reasoning is wrong. It is abusing the properties of convergent sums by applying them to divergent sums. One has to be very careful about definitions before one can blindly apply rules that work within finite systems. I could answer queries about this and other similar results fully if you ask on Maths SE. If you understand the maths, the potential applications will become more apparent to you $\endgroup$
    – user13456
    Jun 2, 2017 at 17:27

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(Don't vote on this; this is just my comment which is too long)

The Riemann-Zeta function, which is a general case for the Ramanujan sum, has many un-intuitive and baffling properties such as:

  • having no closed-form solution
  • not being solvable with linear and stable methods

The first roughly means that you need an infinite number of operations to expand out the actual Riemann-Zeta function. The second roughly means 0 is no longer an additive identity and/or associativity of addition breaks down. These things combined mean that you're gonna have weird issues treating infinite sums as if they were finite sums where you can do EZ-PZ operations with. It's almost as if you have to define the very concept of infinity as two separate, contradicting definitions.

Treating infinite sums as finite sums could be interesting in theoretical topics. But with my money? I don't think my accountant would appreciate me going in and saying that my finite sum of money can be treated as an infinite sum of money.

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