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October 20

Sum of the reciprocals of the subfactorials

What is the sum of the reciprocals of the subfactorials of the integers greater than 1?? (1 subfactorial is 0 and 1/0 is undefined.) Georgia guy (talk) 15:04, 20 October 2016 (UTC)[reply]

Going by derangement, the question is to compute $\sum _{n=2}^{+\infty }{\frac {1}{!n}}=\sum _{n=2}^{+\infty }{\frac {1}{n!\sum _{i=0}^{n}{\frac {(-1)^{i}}{i!}}}}$ . I see no obvious way to attack this. Tigraan^{Click here to contact me} 15:58, 20 October 2016 (UTC)[reply]

If I understand derangement rightly, it says that

!n=\left[{\frac {n!}{e}}\right]

, where that is, bizarrely, a round off to the nearest integer function in brackets. And factorial says that

e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}.

Now I know that we can't say that

\sum _{n=2}^{+\infty }{\frac {e}{n!}}

= e * e¹ minus the first two points, because there's that round off to be considered. But can this be a start, if you understand why this round-off happens and can calculate just the remainders to offset them from this sum??? Wnt (talk) 17:40, 20 October 2016 (UTC)[reply]

I just looked into this a bit further and found [1] which says that the subfactorial is equal to the "uppercase incomplete gamma function" gamma(n+1, -1)/e. Which is really neat, except that I wish they'da named the function something like Blarf234385(tm), because nothing short of at threat of prosecution seems capable of getting two programs to mean the same thing by a function name with a gamma in it. (Well, alright, that's being facetious; blasphemy is the most satisfying form of profanity) I installed R package 'gsl' and used gamma_inc() and variants, but none of them work for negative values of the second parameter. This was kind of off the topic anyway though, since what seems most relevant is that this has the same /e as above. The factorial n! is equal to the complete gamma function, and so it is simply gamma(n+1) = gamma(n+1, 0)... Wnt (talk) 04:15, 21 October 2016 (UTC)[reply]

It converges quickly: I get 1.6382270745053706475428931141511226610635932496444... at n=40. No results found using inverse symbolic calculators. 24.255.17.182 (talk) 04:13, 21 October 2016 (UTC)[reply]