Re: Series

David C. Ullrich wrote:

In article
<12318708.1222361908437.JavaMail.jakarta@xxxxxxxxxxxxx
forum.org>,
Maury Barbato <mauriziobarbato@xxxxxxxx> wrote:

Hello,
let {a_n} be a sequence of real numbers which is
not
in l_2. Is there sum element {x_n} of l_2 such that

sum_{n=0 to inf} (a_n)*(x_n} diverges?

Yes. Hint for an elementary explicit construction:

Partition the positive integers into finite sets
I_1, I_2, ... such that the sum of (a_n)^2 for n in
I_j
is greater than 1. Let c_j = ___________ and then
define x_n = c_j * a_n for n in I_j.

(Hint: if you happened to get the sum of (a_n)^2 for
n in
I_j exactly equal to 1 then c_j = 1/j would work.)

Thank you very much for your attention.
My Best Regards,
Maury

--
David C. Ullrich

I complete the construction. Let s_j be the sum of
(a_n)^2 for n in I_j. Then c_j = 1/(j*s_j) will
work.

Thank you very very .. much, David!
My Best Regards,
Maury
.

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