Sum of reciprocals of double factorials
Jonathan Post
jvospost3 at gmail.com
Wed Aug 6 20:03:07 CEST 2008
Perhaps it is now worth submitting the two sequences of numerators and
denominators of the convergents to this real number, given that Eric
Wesstien has added to MathWorld and OEIS with Max Alekseyev's lovely
formula:
Weisstein, Eric W. "Double Factorial." From MathWorld--A Wolfram Web
Resource. http://mathworld.wolfram.com/DoubleFactorial.html
"The sum of reciprocal double factorials can be given in closed form as
sum_(n=0)^infty1/(n!!)=sqrt(e)[1+sqrt(pi/2)erf(1/2sqrt(2))]"
and
(Sloane's A143280).
This leaves open the analogous:
sum_(n=0)^infty1/(n!!!)
sum_(n=0)^infty1/(n!!!!)
sum_(n=0)^infty1/(n!!!!!)
etcetera, and their convergents.
On Mon, Aug 4, 2008 at 12:41 PM, Jonathan Post <jvospost3 at gmail.com> wrote:
> Very elegant, Max Alekseyev!
>
> The generalization to k-tuple factorials is similar, and the
> convergents require a modulo k adjustment?
>
> In my earlier email, of course I meant "What are the real numbers to
> which the sum of reciprocals of k-tuple
> factorials converges?"
>
> With your formulae, is this now worth submitting, as definitive rather
> than merely an analogue of existing seqs?
>
> Best,
>
> Jonathan Vos Post
>
> On Mon, Aug 4, 2008 at 12:35 PM, Max Alekseyev <maxale at gmail.com> wrote:
>> Corrections:
>>
>> SUM[i=0..oo] 1 / i!! = (sqrt(Pi/2) * erf(1/sqrt(2))+1) * sqrt(e) ~= 3.0594074
>>
>> and
>>
>> to get an expression for a(n) we need to consider the terms on odd and
>> even positions separately:
>>
>> SUM[i=0..oo] x^(2i) / (2i)!! = exp(x^2/2);
>>
>> SUM[i=0..oo] x^(2i+1) / (2i+1)!! = sqrt(Pi/2) * erf(x/sqrt(2)) * exp(x^2/2).
>>
>> Then
>>
>> a(2n) = floor( sqrt(e) * (2n)!! ) / (2n)!! + floor( sqrt(e*Pi/2) *
>> erf(1/sqrt(2)) * (2n-1)!! ) / (2n-1)!!
>> ~= floor( 1.648721271 * (2n)!!) / (2n)!! + floor( 1.410686134 *
>> (2n-1)!! ) / (2n-1)!!
>>
>> and
>>
>> a(2n+1) = floor( sqrt(e) * (2n)!! ) / (2n)!! + floor( sqrt(e*Pi/2) *
>> erf(1/sqrt(2)) * (2n+1)!! ) / (2n+1)!!
>> ~= floor( 1.648721271 * (2n)!!) / (2n)!! + floor( 1.410686134 *
>> (2n+1)!! ) / (2n+1)!!
>>
>> Regards,
>> Max
>>
>> On Mon, Aug 4, 2008 at 12:26 PM, Max Alekseyev <maxale at gmail.com> wrote:
>>> Note that
>>>
>>> SUM[i=0..oo] x^i / i!! = (sqrt(Pi/2) * erf(x/sqrt(2))+1) * exp(x^2/2)
>>>
>>> so that
>>>
>>> SUM[i=0..oo] x^i / i!! = (sqrt(Pi/2) * erf(1/sqrt(2))+1) * sqrt(e) ~= 3.0594074
>>>
>>> and your sequence can be defined as
>>>
>>> a(n) = floor( (sqrt(Pi/2) * erf(1/sqrt(2))+1) * sqrt(e) * n!! ) / n!!
>>> ~= floor( 3.0594074 * n!! ) / n!!
>>>
>>> Regards,
>>> Max
>>>
>>> On Mon, Aug 4, 2008 at 11:55 AM, Jonathan Post <jvospost3 at gmail.com> wrote:
>>>> Is this worth adding to OEIS?
>>>>
>>>> Sum of reciprocals of double factorials
>>>> (2 seqs "frac" for numerators and denominators)
>>>>
>>>> SUM[i=0..n] 1/A006882(i)
>>>>
>>>> n numerator/denominator
>>>> 0 1/0!! = 1/1
>>>> 1 1/0!! + 1/1!! = 2/1
>>>> 2 1/0!! + 1/1!! + 1/2!! = 5/2
>>>> 3 1/0!! + 1/1!! + 1/2!! + 1/3!! = 17/6
>>>> 4 1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! = 71/24
>>>> 5 1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! + 1/5!! = 121/40
>>>> 6 1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! + 1/5!! + 1/6!! = 731/240
>>>> 7 1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! + 1/5!! + 1/6!! + 1/7!! =
>>>> 1711/560 ~ 3.0553571
>>>> 8 1/0!! + 1/1!! + 1/2!! + 1/3!! + 1/4!! + 1/5!! + 1/6!! + 1/7!! +
>>>> 1/8!! = 41099/13440 ~ 3.05796131
>>>>
>>>> The series obviously converges (being of order 1/n^2).
>>>>
>>>> This is to double factorials A006882 as A007676/A007677is to factorial.
>>>>
>>>> The WIMS continued fraction online calculator seems to be unavailable
>>>> at the moment, O i've stopped with the above by-hand draft.
>>>>
>>>> If this is of interest, then there would be an array A[k,n] = nth
>>>> convergent to sum of reciprocals of the k-th multiple factorial, using
>>>> the correct definitions by njas, Robert G. Wilson v, Mira Bernstein of
>>>> k-th multiple factorial.
>>>>
>>>> What is the real number to which the sum of reciprocals of double
>>>> factorials converges?
>>>>
>>>> What are the real numbers to which the sum of reciprocals of double
>>>> factorials converges?
>>>>
>>>
>>
>
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