How Can Divergent Series Be Resumed?

• eljose79
In summary, a divergent series is a mathematical series with no finite limit as the number of terms increases. Resumming such series is important because it assigns a meaningful value to them and allows for practical applications. This is done through various methods such as Borel resummation and zeta function regularization, each with its own advantages and limitations. The resummation of divergent series has many applications in mathematics and science, but also faces challenges and controversies in terms of method choice and disagreement among experts.
eljose79
How is this made?..in fact from having infinites you sum them and have a finite number...i do not know how you can do it..what techniques of maths are used.and if this would be valid for making any series convergent or have a finite number...i think you use divergent series math theories..could someone explain me how to resumate divergent series?..thanks.

I wrote a brief note about it

http://dftuz.unizar.es/~rivero/research/0208180.pdf

Key word in this matter is Borel resummation.

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1. What is a divergent series?

A divergent series is a mathematical series whose terms do not approach a finite limit as the number of terms increases. This means that the sum of the series does not have a definite value and can potentially go to infinity.

2. Why is the resummation of divergent series important?

The resummation of divergent series is important because it allows us to assign a meaningful value to a series that would otherwise be considered meaningless or infinite. It allows us to make sense of divergent series in various mathematical and scientific contexts.

3. How is the resummation of divergent series done?

There are various methods for resumming divergent series, including Borel resummation, zeta function regularization, and analytic continuation. Each method has its own advantages and limitations, and the choice of method depends on the specific series and its intended application.

4. What are some applications of resumming divergent series?

The resummation of divergent series has applications in many areas of mathematics and science, including quantum field theory, statistical mechanics, and number theory. It is also used in practical applications such as signal processing and data analysis.

5. Are there any challenges or controversies surrounding the resummation of divergent series?

Yes, there are ongoing debates and controversies surrounding the resummation of divergent series, especially when it comes to choosing the most appropriate method for a given series. Some methods may produce different results or may not work for certain types of series, leading to disagreement among mathematicians and scientists.

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