({{, (RefLink[Regularization, paclet:ref/Regularization]) 
is an option for (RefLink[Sum, paclet:ref/Sum]) and (RefLink[Product, paclet:ref/Product]) that specifies what type of regularization to use.}})

OpenersNotesSection(Spacer1)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 (∑_(n==0)^∞)f(n):

For alternating sums (∑_(n=0)^∞)(((-1))^n) f(n), the setting "Euler" gives (∑_(p==0)^∞)(((-1))^p)(( (Subscript[((Δ^p) f(k)), k==0]))/(2^(p+1))).

The following setting can be used to specify a regularization procedure for products (∏_(n==1)^∞)f(n):

(RefLink[Regularization, paclet:ref/Regularization])->(RefLink[None, paclet:ref/None]) specifies that no regularization should be used.

For multiple sums and products, the same regularization is by default used for each variable.

(RefLink[Regularization, paclet:ref/Regularization])->{(Subscript[reg, 1]),(Subscript[reg, 2]),…} specifies regularization (Subscript[reg, i]) for the i(^th) variable.

OpenersPrimaryExamplesSection(Spacer1)Examples  (9)

OpenersExampleSection0(Spacer1)Basic Examples  (3)

The following sum does not converge:

Sum[(-1)^n,{n,0,Infinity}]

Using Abel regularization will produce a finite value:

Sum[(-1)^n,{n,0,Infinity},Regularization->"Abel"]

In this case the Abel-regularized sum does not exist:

Sum[(-2)^n,{n,0,Infinity},Regularization->"Abel"]

However, the stronger Borel regularization produces a finite value:

Sum[(-2)^n,{n,0,Infinity},Regularization->"Borel"]

A regularized value of a divergent product:

Product[k^2,{k,1,Infinity},Regularization->"Dirichlet"]

OpenersExampleSection1(Spacer1)Scope  (5)

Apply Abel regularization to sum a divergent polynomial-exponential series:

Sum[(((-1))^n) (((n+1))^3),{n,0,∞},Regularization->"Abel"]

Use Borel regularization to sum a divergent hypergeometric series:

Sum[((Gamma[3 n+(1/2)] (((-1))^n) (((((2 (z^(3/2)))/3)))^(-n)))/((2^n) (3^(3 n)) Gamma[n+(1/2)] Gamma[n+1])),{n,0,∞},Regularization->"Borel"]

Apply Cesaro regularization to sum a divergent trigonometric series:

Sum[Sin[n],{n,0,∞},Regularization->"Cesaro"]

Sum a divergent logarithmic series using Dirichlet regularization:

Sum[Log[n],{n,1,∞},Regularization->"Dirichlet"]

Apply Euler regularization to sum a divergent geometric series:

Sum[(((-3))^n),{n,0,∞},Regularization->"Euler"]

OpenersExampleSection3(Spacer1)Applications  (1)

The regularized sum of all the natural numbers is -1/12:

Sum[n,{n,1,∞},Regularization->"Dirichlet"]

Zeta[-1]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[SumConvergence, paclet:ref/SumConvergence])  ▪  (RefLink[VerifyConvergence, paclet:ref/VerifyConvergence])  ▪  (RefLink[GenerateConditions, paclet:ref/GenerateConditions])}})}})