log(m)'s close to integers (sum convergence)
Leroy Quet
qqquet at mindspring.com
Mon Mar 17 01:55:13 CET 2003
I am afraid that this question might either be trivial or already
answered, in a way, a little while back when this group was discussing
this sequence.
Consider the sequence:
>ID Number: A080021
>Sequence: 2,3,7,20,148,403,1096,1097,2980,2981,8103,59874,162755,
> 442413,1202604,3269017,8886110,8886111,24154952,24154953,
> 65659969,178482301,3584912846,9744803446,26489122130,
> 72004899337,195729609428
>Name: log(n) is closer to an integer than is log(m) for any m with
> 2<=m<n.
>Comments: Every term is floor(e^k)+r for some integers k and r with k>=1
>and -1
> <= r <= 1.
>References Suggested by Leroy Quet (qqquet at mindspring.com), Jan 19 2003
>Example: log(2) = 1-0.306..., log(3) = 1+0.0986..., log(7) = 2-0.0540...,
> log(20) = 3-0.00426...
>See also: Cf. A080022, A080023.
>Keywords: nonn
>Offset: 0
>Author(s): Dean Hickerson (dean at math.ucdavis.edu), Jan 20 2003
>Extension: More terms from Don Reble (djr at nk.ca), Jan 20 2003
Is the sum of the reciprocals of all of the terms (in the above sequence)
convergent?
ie. Does sum{k=0 to oo} 1/a(k) converge?
(There are many terms close to each other, so the sum may very well
diverge.)
Thanks,
Leroy Quet
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