This is documentation for Mathematica 7, which was
based on an earlier version of the Wolfram Language.
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Regularization

Regularization
is an option for Sum and Product that specifies what type of regularization to use.
  • Regularization affects only results for divergent sums and products.
  • The following settings can be used to specify regularization procedures for sums of the form sum_(n=0)^inftyf(n):
"Abel"
"Borel"
"Cesaro"
"Dirichlet"
  • For alternating sums sum_(n=0)^infty(-1)^n f(n), the setting "Euler" gives .
  • The following setting can be used to specify a regularization procedure for products product_(n=0)^inftyf(n):
"Dirichlet"
  • 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.
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:
The following sum does not converge:
In[1]:=
Click for copyable input
Out[1]=
Using Abel regularization will produce a finite value:
In[2]:=
Click for copyable input
Out[2]=
 
In this case the Abel regularized sum does not exist:
In[1]:=
Click for copyable input
Out[1]=
However, the stronger Borel regularization produces a finite value:
In[2]:=
Click for copyable input
Out[2]=
 
A regularized value of a divergent product:
In[1]:=
Click for copyable input
Out[1]=
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