# Sum of reciprocals of double factorials

Max Alekseyev maxale at gmail.com
Mon Aug 4 21:26:33 CEST 2008

```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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