compute closed forms of definite and indefinite sums - Maple Help

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SumTools[Summation] - compute closed forms of definite and indefinite sums

SumTools[DefiniteSummation] - compute closed forms of definite sums

SumTools[IndefiniteSummation] - compute closed forms of indefinite sums

Calling Sequence

Summation(f, k)

Summation(f, k=m..n, parametric)

Summation(f, k=alpha)

Summation(f, expr)

DefiniteSummation(f, k=m..n, parametric)

DefiniteSummation(f, k=alpha)

DefiniteSummation(f, k=expr)

IndefiniteSummation(f, k)

Parameters

f

-

expression; specified summand

k

-

name; summation index

m, n

-

integers or expressions

parametric

-

(optional) literal name

alpha

-

RootOf expression

expr

-

expression not containing k

Description

• 

The IndefiniteSummation(f, k) command computes a closed form of the indefinite sum of f with respect to k, that is, it finds a function gk such that gk+1gk=fk.

• 

The DefiniteSummation(f, k=m..n) command computes a closed form of the definite sum of fk over the specified range m..n.

  

For notational convenience, there are two special cases:

  

- The DefiniteSummation(f, k=alpha) command, where alpha is a RootOf structure, computes the definite sum of fk over the index of the specified RootOf

  

- The DefiniteSummation(f, k=expr) command is used for function evaluation at a specified point. This is an equivalence of eval(f, k=expr).

• 

The Summation(f, ...) command is the combination of DefiniteSummation and IndefiniteSummation commands.

  

For a specified summation problem give to Summation, there is a preprocess that classifies whether the summation is a definite or indefinite sum.  It then calls either IndefiniteSummation or DefiniteSummation as appropriate.

• 

If Summation, IndefiniteSummation, or DefiniteSummation fails to compute a closed form, the unevaluated sum returns.

  

Note: The sum routine recognizes various resummation methods and thus is able to give the 'correct' value for various classes of divergent sums. If the environment variable _EnvFormal is set to true, sum uses this technique.

  

If _EnvFormal is set to false, then Maple tries harder to determine whether the given definite sum is divergent and returns infinity, -infinity, or unevaluated in that case. Note that this may slow down the computation.

  

In that case, Maple also tries to recognize removable singularities in the summation interval.

  

By default, _EnvFormal is unassigned.

• 

For more information, see sum,details.

Examples

withSumTools:

Two examples of indefinite sums.

Example 1.

F:=binomialn2+k,k2k

F:=binomial12n+k,k2k

(1)

kF=IndefiniteSummationF,n

kbinomial12n+k,k2k=12nbinomial12n+k,k2kk+1+12n+1binomial12n+12+k,k2kk+1

(2)

Example 2.

F:=n2binomial2n,nn2+3n+2

F:=n2binomial2n,nn2+3n+2

(3)

nF=IndefiniteSummationF,n

nn2binomial2n,nn2+3n+2=42n1_i=1n112_i+12_i+1n+1n+nn214n+8_i=1n1_i4_i+22n+2

(4)

Two examples of definite sums:

Example 1.

F:=binomial2n+1,2k2

F:=binomial2n+1,2k2

(5)

k=0nF=DefiniteSummationF,k=0..n

k=0nbinomial2n+1,2k2=2Γ2n+12n4n+22n322Γ2n+12Γn+1Γn+32

(6)

Example 2.

F:=22kΓknΓk+nzkπΓ2k+1

F:=22kΓn+kΓk+nzkπΓ2k+1

(7)

k=0∞F=DefiniteSummationF,k=0..∞

k=0∞22kΓn+kΓk+nzkπΓ2k+1=πcos2narcsinzcscπnn

(8)

Summationk32,k=1..∞

∞

(9)

Summation1k,k=1..∞

k=1∞1k

(10)

Parametric case discussions may be returned:

F:=binomial2k3,k4k

F:=binomial2k3,k4k

(11)

k=0nF=SummationF,k=0..n,parametric

k=0nbinomial2k3,k4k={0n=12n+1n+2binomial2n1,n+1n14n+1n034n=12n+1n+2binomial2n1,n+1n14n+1+382n

(12)

Summation1k,k=a..b

Warning, unable to determine if the summand is singular in the interval of summation; try to use assumptions or use the parametric option

k=ab1k

(13)

Summation1k,k=a..b,parametric

{0a=b+1Ψb+1Ψa1aand0bΨbΨ1aa0andb1FAILotherwise

(14)

If _EnvFormal is set to true, the Summation command returns the class for recognized divergent sums.

_EnvFormal:=true

_EnvFormal:=true

(15)

Summationk32,k=1..∞

ζ32

(16)

Summation1k,k=1..∞

12

(17)

See Also

product, RootOf, sum, SumTools


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