Cont Frac of Harmonic Numbers
benoit
abcloitre at wanadoo.fr
Sat Dec 20 09:02:17 CET 2003
> http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/
> eisA.cgi?A
> num=A055573
>
> "lim n -> infinity a(n)/n = C = 0.84... "
>
> How is it proved that C exists? What are more digits of C? And, of
> course, does C have a closed-form?
>
That was experimental result :
For m integer >=1 let a(n,m)= number of elements in continued fraction
for sum(i=1,n,1/i^m)
I noticed that for fixed m, limit n-->infty a(n,m)/n exists = C(m)
(cf. A070985)
I noticed also C(m) is closed from m*C(1)=m*C. Does C(m)=m*C ?
If true, constant C=0.84... would be more interesting.
Some values :
m----a(10000,m)/(10000*m)
1-----0.835599
2-----0.841299
3-----0.838599
4-----0.845199
5-----0.843979
6-----0.839083
7-----0.849357
8-----0.838137
Seems support the conjecture.
> And finally,
> what are the asymptotics of the related sequence:
>
> http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/
> eisA.cgi?A
> num=A058027
>
>
Should be asymptotic to D*n^(3/2) D=0.4....
BC
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