Where is sum of divergent series and values of divergent integrals used in physics?

Question

Hello everyone!

My question is: 

Where is sum of divergent series and divergent integrals used in physics? What it all means? Where can I find examples of divergent integrals? Is there a book of problems for physicists? 

I am mathematician. I developed a method for summing divergent series and determines the value of divergent integrals https://m4t3m4t1k4.wordpress.com/2015/02/14/general-method-for-summing-divergent-series-determination-of-limits-of-divergent-sequences-and-functions-in-singular-points-v2/ I am fascinated by the time they have the application in physics. What is the point of it all? How to interpret all physicists? Also looking for examples of divergent integrals to challenge that method. So far I've found three examples that I have tested in the work. 

Thanx.

Comments

Cross-posted to Physics.SE and PhysicsForums.

The paragraph linking to some blog post doesn't seem relevant to the rest of the post (and probably a bad attempt at self-promotion).

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This is undergraduate-level, voting to close.

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Voting to close.

Answer 1

Path integrals are an obvious example - perturbation series are often divergent sums. Renormalisation is all about getting rid of these infinities. For example, the Ramanujan sum is encountered rather frequently in QFT and string theory, \(\varepsilon = \sum_{n=0}^\infty n := -1/12\), represents the Casimir energy in string theory, for example.

Comments

Thanx dimension10. Where can I find examples of divergent integrals? Is there a book of problems for physicists?

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@sinisa Do a simple web search, keywords "perturbation series", "renormalisation", "ramanujan sums qft", etc. There are lots of them.