Regularization
Details
- Regularization affects only results for divergent sums and products.
- The following settings can be used to specify regularization procedures for sums of the form :
-
"Abel""Borel""Cesaro""Dirichlet"
- For alternating sums , the setting "Euler" gives .
- The following setting can be used to specify a regularization procedure for products :
-
"Dirichlet"
- Regularization->None specifies that no regularization should be used.
- For multiple sums and products, the same regularization is by default used for each variable.
- Regularization->{reg1,reg2,…} specifies regularization regi for the i^(th) variable.
Examples
open all close allBasic Examples (3)
The following sum does not converge:
Using Abel regularization will produce a finite value:
In this case the Abel-regularized sum does not exist:
However, the stronger Borel regularization produces a finite value:
A regularized value of a divergent product:
Scope (5)
Apply Abel regularization to sum a divergent polynomial-exponential series:
Use Borel regularization to sum a divergent hypergeometric series:
Apply Cesaro regularization to sum a divergent trigonometric series:
Sum a divergent logarithmic series using Dirichlet regularization:
Apply Euler regularization to sum a divergent geometric series:
Applications (1)
The regularized sum of all the natural numbers is :
History
Text
Wolfram Research (2008), Regularization, Wolfram Language function, https://reference.wolfram.com/language/ref/Regularization.html.
CMS
Wolfram Language. 2008. "Regularization." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Regularization.html.
APA
Wolfram Language. (2008). Regularization. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Regularization.html
BibTeX
@misc{reference.wolfram_2025_regularization, author="Wolfram Research", title="{Regularization}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/Regularization.html}", note=[Accessed: 16-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_regularization, organization={Wolfram Research}, title={Regularization}, year={2008}, url={https://reference.wolfram.com/language/ref/Regularization.html}, note=[Accessed: 16-November-2025]}