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I am completely fascinated by the Euler/Ramanujan result

$$ \sum_{n=0}^{\infty} n = -\frac{1}{12}$$

It is amazing to me that there are so many seemingly bogus ways to evaluate this, and they all give the same ridiculous answer. It's magical. I want to understand what's going on here more deeply.

I have been reading up on the Euler-MacLaurin formula, Ramanujan Summation, and the Abel-Plana formula, but wikipedia does not go into much depth. Where can I find what I need to understand and prove these results? I have an undergrad degree in math, but still get quickly lost in most research-level papers.

luqui
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    I haven't read it carefully, but Terry Tao had a blog post about it once. https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/ – Aaron Feb 04 '19 at 04:09

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