- #1
tpm
- 67
- 0
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' )
*BOREL
-------
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
--------------
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)
---------
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' )
