Criticism about resummation methods.

[tpm]

I have been studying the 'resummation' methods for divergent series..however i have some questions of critcs.

*BOREL
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Yes Borel method is very beatiful however ..can you get for every sequence of a(n) so $$\sum_{n=0}^{\infty} a_{n}$$ is divergent, the value of:

$$\sum_{n=0}^{\infty} a_{n}\frac{x^{n}}{n!} =f(x)$$ ??

* RIESZ MEAN
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You have the same problem, for example for 'lambda' big you will never be able to give a value for expressions like:

$$\sum_{n \le \lambda}(1- \frac{n}{\lambda})^{\delta} \Lambda (n)$$


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Hence, my opinion is that if we can only use this resummation method for only a few cases are they still useful ?? (i'm not saying these methods are WORNG but perhaps they are completely 'useless' )

 

[fresh_42]

You must not look from the end of the process, but from the beginning. Those are tools and as such serve a purpose. They don't make and do not need to make much sense without a goal. "Why would someone want to sum divergent series?" has to be answered first.

https://en.wikipedia.org/wiki/Borel_summation
https://en.wikipedia.org/wiki/Riesz_mean