Undergrad Understanding the summation of diverging series

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SUMMARY

The discussion centers on the summation of diverging series, specifically addressing the claim that the sum of all natural numbers equals -1/12. This assertion is invalid; the sum of all natural numbers is infinity. The conversation highlights various summation methods, including Abel summation, Cesàro summation, and Ramanujan summation, which can assign finite values to divergent series, but these values do not represent the actual sums of the series. The conclusion emphasizes that while Ramanujan summation can yield -1/12, it does not change the fact that the sum of natural numbers diverges to infinity.

PREREQUISITES
  • Understanding of divergent series and their properties
  • Familiarity with summation methods such as Abel and Cesàro summation
  • Basic knowledge of Ramanujan summation techniques
  • Concepts in string theory and critical dimensions
NEXT STEPS
  • Research the principles of Abel summation and its applications
  • Explore Cesàro summation and its relevance in mathematical analysis
  • Study Ramanujan summation and its implications in theoretical physics
  • Investigate the role of divergent series in string theory and critical dimensions
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Mathematicians, physicists, and students interested in advanced mathematical concepts, particularly those exploring the implications of divergent series in theoretical frameworks like string theory.

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I was recently researching into some string theory when i came across the following summation:
The sum of all natural numbers is -1/12, now I'm still wrapping my head around the context of the application within critical string dimensions, but is this summation valid? And if not, why it being used in this way?
 
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No, it's not valid. The sum of all natural numbers is infinity, and not -1/12. There are certain ways to assign a number to divergent series however, those ways are called summation methods. You have Abel summation, Cesaro summation, Ramanujan summation, etc. These numbers are not the sums of the series in question. So while ##1+2+3+4+...## can be Ramanujan summed to -1/12, the sum of all natural numbers is still infinity, as it should be.
 
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