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From: Andreas on 5 May 2010 03:56
Given the real sequence x_n = 1 if n = 3, 3^2, 3^3,... and x_n = 0 otherwise what is the asymptotics for sum(x_k, k=1..n)? At least can be affirmed sum(x_k, k=1..n) = O(log n), n -> oo? (Big O)

Regards,
Andreas U. Kelmer
From: Torsten Hennig on 5 May 2010 04:05
> Given the real sequence x_n = 1 if n = 3, 3^2,
> 3^3,... and x_n = 0 otherwise what is the asymptotics
> for sum(x_k, k=1..n)? At least can be affirmed
> sum(x_k, k=1..n) = O(log n), n -> oo? (Big O)
>
> Regards,
> Andreas U. Kelmer

Isn't
sum_{k=1}^{n} x_k = [log_{3} n]
where [x] is the biggest integer less or equal to x ?

Best wishes
Torsten.
From: Ray Vickson on 5 May 2010 11:16
On May 5, 5:05 am, Torsten Hennig <Torsten.Hen...(a)umsicht.fhg.de>
wrote:
> > Given the real sequence x_n = 1 if n = 3, 3^2,
> > 3^3,... and x_n = 0 otherwise what is the asymptotics
> > for sum(x_k, k=1..n)? At least can be affirmed
> > sum(x_k, k=1..n) = O(log n), n -> oo? (Big O)
>
> > Regards,
> > Andreas U. Kelmer
>
> Isn't
> sum_{k=1}^{n} x_k = [log_{3} n]
> where [x] is the biggest integer less or equal to x ?

No. [log_{3}(n)] = 2 for n = 3^2, 3^2+1, 3^2+2, ..., 3^3-1, but the OP
wants x_n = 0 for n = 3^2+1, 3^2+2,..., 3^3-1.

R.G. Vickson

>
> Best wishes
> Torsten.

From: Rob Johnson on 5 May 2010 11:30
In article <5e08ba9f-586f-4290-837a-d168df7c4970(a)t26g2000prt.googlegroups.com>,
Ray Vickson <RGVickson(a)shaw.ca> wrote:
>On May 5, 5:05 am, Torsten Hennig <Torsten.Hen...(a)umsicht.fhg.de>
>wrote:
>> > Given the real sequence x_n = 1 if n = 3, 3^2,
>> > 3^3,... and x_n = 0 otherwise what is the asymptotics
>> > for sum(x_k, k=1..n)? At least can be affirmed
>> > sum(x_k, k=1..n) = O(log n), n -> oo? (Big O)
>>
>> > Regards,
>> > Andreas U. Kelmer
>>
>> Isn't
>> sum_{k=1}^{n} x_k = [log_{3} n]
>> where [x] is the biggest integer less or equal to x ?
>
>No. [log_{3}(n)] = 2 for n = 3^2, 3^2+1, 3^2+2, ..., 3^3-1, but the OP
>wants x_n = 0 for n = 3^2+1, 3^2+2,..., 3^3-1.

Actually, the OP wanted the sum of x_n, and that does seem to be
[log_3(n)].

Thus, the infimum of the constants in the OP's big-O estimate is
1/log(3).

Rob Johnson <rob(a)trash.whim.org>
take out the trash before replying
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