Proof for equation (1)pic.twitter.com/byTLCNaAei
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I want to be clear. I do not assert any "equality" of the sum to -1/12 in the limit that ε-->0. This makes no sense What does make sense is to speak about a "finite piece" and a divergent piece. This result is unique, as outlined by Terrance Tao.https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/ …
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Sometimes in physics, absolute quantities don't have a meaning. Rather what does have a meaning is *differences* . In these situations, the term that *would* diverge cancels out completely leaving behind a finite answer; giving way to the following "replacement rule"pic.twitter.com/QHm3LbXoai
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I've had a lot of requests to post about a physical application of the replacement ∑n → -1/12. So without further ado I give you: The Casimir Effect
pic.twitter.com/xsxVAjtxEA
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Quick question. Do you type in all the raw LaTeX code or use some kind of wsywg editor. If so, what program are you using?
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Hi mister, if I am not mistaken, it is dangerous to play with infinite series in this way - unless you are joking. It is there are several unsafe steps in this pic.
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3. In the "equation 2" part, the leading term of 1/epsilon^2 is clearly exploding to infinity, while the large blob on the right of it is 1.
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The RHS of eq (3) diverges when ε⇒0
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Right but that’s the point.. I’m not taking the limit.. everything is finite until the very last step where the divergence is clearly isolated to one term
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