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SumTools[IndefiniteSum]

  

Indefinite

  

compute closed forms of indefinite sums

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

Indefinite(f, k, opt)

Parameters

f

-

expression depending on k

k

-

name

opt

-

(optional) equation of the form failpoints=true or failpoints=false

Description

• 

The Indefinite(f, k) command computes a closed form of the indefinite sum of f with respect to k.

• 

The command uses a combination of algorithms handling various classes of summands. They include the classes of polynomials, rational functions, and hypergeometric terms and j-fold hypergeometric terms, functions for which their minimal annihilators can be constructed, for example, d'Alembertian terms. For more information, see LinearOperators.

  

A library extension mechanism is also used to include sums for which an algorithmic approach to finding closed forms does not yet exist. Currently the computable summands include expressions containing the harmonic function, logarithmic function, digamma and polygamma functions, and sin, cos, and the exponential functions.

• 

If the option failpoints=true (or just failpoints for short) is specified, then the command returns a pair s,p,q, where

– 

s is a closed form of the indefinite sum of f w.r.t. k, as above,

– 

p is a list of intervals a1..b1,a2..b2,...,am..bm where f does not exist, and

– 

q is a list of points ki where the computed sum s does not satisfy the telescoping equation ski+1ski=fki or does not exist.

  

If such points appear in the summation interval, the discrete Newton-Leibniz formula may fail.

• 

If the command is unable to compute one of the lists p,q, it returns s,FAIL.

Examples

withSumTools[IndefiniteSum]:

An example of a rationally summable expression:

f1n2+5n1

f:=1n2+5n1

(1)

sIndefinitef,n

s:=13n32+12513n12+12513n+12+125

(2)

Check the telescoping equation:

evalaNormalsn=n+1|sn=n+1s,expanded

1n2+5n1

(3)

A hypergeometrically summable term:

f22n1n2n+1binomial2n,n

f:=22n1n2n+1binomial2n,n

(4)

sIndefinitef,n

s:=22n1binomial2n,nn

(5)

normalexpandsn=n+1|sn=n+1s

122n2n2n+1binomial2n,n

(6)

The method of accurate summation:

f112+15122n1215122n25

f:=1512+125n12125n2

(7)

Indefinitef,n

31012+125n12125n211012+125n+112125n+12+11012+125n+212125n+22

(8)

Sum of a logarithm of a rational function (provided the argument of the logarithm has constant sign):

nln2n+1assuming0n

nln2+lnΓ12+n

(9)

nln2n+1assumingn1

nln2lnΓ12n+Inπ

(10)

Example for the library extension mechanism:

fsinncosn+1Ψn

f:=sinncosn+1Ψn

(11)

Indefinitef,n

12sin12n+2Ψn+1nsin1+cosn22γsin12Ψnsin12sin1nsin1

(12)

Compute the fail points:

fnn!

f:=nn!

(13)

Indefinitef,n,'failpoints'

n!,∞..1,

(14)

Indeed, f is not defined for any negative integer:

fn=3|fn=3

Error, numeric exception: division by zero

and limits do not exist:

limn→3f

undefined

(15)

A rational example. f and its limit are not defined at n=0,1,5, and the correspondent sum s and its limit are not defined at n=2,3,4:

f1n+1n12n5

f:=1n+1n12n5

(16)

s,fpIndefinitef,n,'failpoints'

s,fp:=2n5+2n4+2n3+2n2+1n1,0..1,5..5,2,3,4

(17)

limn→2s

undefined

(18)

In the next example, f is hypergeometric term defined for all integers n:

fbinomial2n3,n4n

f:=binomial2n3,n4n

(19)

s,fpIndefinitef,n,'failpoints'

s,fp:=2nn+1binomial2n3,nn24n,,2

(20)

The sum s is not defined at n=2:

sn=2|sn=2

Error, numeric exception: division by zero

Note that in this example, however, the limit exists:

limn→2s

38

(21)

but the telescoping equation does not hold at n=1:

limn→2ssn=1|sn=1=fn=1|fn=1

58=14

(22)

Consequently, if n=1 is between summation bounds, the Newton-Leibniz formula is wrong:

n=010f=sn=11|sn=11sn=0|sn=0

236871262144=138567262144

(23)

Rewriting f in terms of GAMMA functions introduces additional singularities at negative integers. These singularities are removable:

f1convertf,Γ

f1:=Γ2n2Γn2Γn+14n

(24)

s,fpIndefinitef1,n,'failpoints'

s,fp:=2nn+1Γ2n2n2Γn2Γn+14n,,

(25)

The telescoping equation is valid for all integers n (in the limit):

n=010f1=sn=11|sn=11limn→0s

138567262144=138567262144

(26)

The singularities of f1 are detected if _EnvFormal (see sum,details) is set to false:

_EnvFormalfalse:

s,fpIndefinitef1,n,'failpoints'

s,fp:=2nn+1Γ2n2n2Γn2Γn+14n,∞..2,

(27)

n=010f1

Error, (in SumTools:-DefiniteSum:-ClosedForm) summand is singular in the interval of summation

_EnvFormal'_EnvFormal':

See Also

SumTools[DefiniteSummation]

SumTools[IndefiniteSum]

SumTools[IndefiniteSum][Hypergeometric]

SumTools[Summation]

 


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