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From: Traxler on 22 Feb 2010 23:36
Hi,

Suppose a(n)>0 is a periodic sequence with:
(a(1)+a(2)+..+a(n))/n--> M
as n-->infty.

Let b(n)>0 be a nonperiodic bounded sequence.

I need to have:

(a(1)b(1)+..+a(n)b(n))/(b(1)+..+b(n))-->M
as n-->infty.

But I guess one needs extra conditions for b sequence.

Thanks for help.

B
From: Dan Cass on 24 Feb 2010 04:24
> Hi,
>
> Suppose a(n)>0 is a periodic sequence with:
> (a(1)+a(2)+..+a(n))/n--> M
> as n-->infty.
>
> Let b(n)>0 be a nonperiodic bounded sequence.
>
> I need to have:
>
> (a(1)b(1)+..+a(n)b(n))/(b(1)+..+b(n))-->M
> as n-->infty.
>
> But I guess one needs extra conditions for b
> sequence.
>
> Thanks for help.
>
> B

Suppose the a sequence goes x,y,x,y,... with x,y different.

It seems that then one can arrange the b's in a
nonperiodic fashion as follows:

Let the b's which multiply the x's all be 0,
while those multiplying the y's are either 0 or 1 in
some (any) nonperiodic fashion.


Then it would seem that
[1] a(1)+a(2)+..+a(n))/n--> M = (x+y)/2,
and yet
[2] (a(1)b(1)+..+a(n)b(n))/(b(1)+..+b(n))--> x not M.
From: Traxler on 25 Feb 2010 01:03
Thanks Dan. sci math research is ok now, so this thread is continued elsewhere.
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