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### What is the relation between renormalization in physics and divergent series in mathematics?

The theory of Divergent Series was developed by Hardy and other mathematicians in the first half of the past century, giving rigorous methods of summation to get unique and consistent results from ...
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### How does the sum of the series “1 + 2 + 3 + 4 + 5 + 6…” to infinity = “-1/12”? [duplicate]

How does the sum of the series “1 + 2 + 3 + 4 + 5 + 6…” to infinity = “-1/12”, in the context of physics? I heard Lawrence Krauss say this once during a debate with Hamza Tzortzis (http://youtu.be/...
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### Critical Dimension of Bosonic Strings and Regularization of $\sum_{n=1}^\infty n$

If $D$ is critical dimension of Bosonic strings, a particular derivation goes like the following, where we arrive finally at $$\frac{D-2}{2}\sum_{n=1}^\infty n + 1 = 0.$$ Now mathematically this is ...
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I read from some books of number theory that $$\sum_{n=1}^{\infty}\frac{1}{n^s} = -\frac{1}{12}\text{,when } s=-1.$$ Now is there such a result $$\sum_{n=1}^{\infty}\frac{1}{n^s} = \pi \text{,when } ... 2answers 435 views ### What is the reason/significance of using  \sum\limits_{n=1}^{\infty}n\rightarrow-\frac{1}{12}? What is the reason/significance of using a trick equation in the Volume I - String Theory - Joseph Polchinsky? I have no doubts at all that the author knows extremely well the subject and that this ... 2answers 893 views ### Is the fact that the sum of all natural numbers \sum_{n=1}^\infty n = -\frac{1}{12} essential to the understanding of the Casimir Force In QED? Apparently this result is used in many areas physics including the extra dimensions of string theory, which is not the scope of the question. The result is apparently also used to understand the ... 1answer 440 views ### Divergent sum in lightcone quantization of bosonic string theory I had the following question regarding lightcone quantization of bosonic strings - The normal ordering requirement of quantization gives us this infinite sum \sum_{n=1}^\infty n. This is regularized ... 1answer 466 views ### Applications of 1+2+3+… = -1/12 The equation$$ 1 + 2 + 3 + \dots = -1/12  is quite famous. From the point of view of mathematics, I have no problem with it. My (probably naive) understanding is that there are certain "sums'' ...
I watched a video of numberphile in which they explain that how you can get Grandi's series sum as $1/2$ ( by Cesàro summation). Then they also give one example of flipping of bulb $1$ means turn on ...