What is up with people thinking that 1 + 2 + 3 + = -1/12?

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The claim that the sum 1 + 2 + 3 + ... equals -1/12 is based on controversial mathematical manipulations, often criticized for being invalid. These manipulations are linked to a technique called Zeta regularization, which is used in theoretical physics, particularly in String Theory. Traditional definitions of convergence, as taught in Real Analysis, show that the series is unbounded and does not converge to a finite value. Despite the mathematical issues, the concept has found applications in physics. Overall, the discussion highlights a divide between rigorous mathematical standards and their use in theoretical physics.
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What is up with people thinking that 1 + 2 + 3 + ... = -1/12?

My friend showed me a Youtube video where a Physicist "proves" that ∑kεNk = -1/12. The guy in the video uses a lot of illegal maneuvers. But apparently this "fact" is used in String Theory. Do Physicists use different definitions of convergence or something? I can easily prove that the sequence {1 + 2 + ... + n} is unbounded using the definitions I learned in Intro to Real Analysis in college.
 
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We had a recent thread on this.

tl;dr The manipulations are indeed invalid. But physicists do use the Zeta regularization in their work.
 

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