Hoi,

I was wondering if the following was true. I can't prove it...but it looks true

suppose

for series in

Then

Sounds like something that should be trivially true..but i have hard time showing it..

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- October 14th 2012, 11:01 AMDinkydoeShow that absolute sum converges
Hoi,

I was wondering if the following was true. I can't prove it...but it looks true

suppose

for series in

Then

Sounds like something that should be trivially true..but i have hard time showing it.. - October 14th 2012, 11:18 AMFernandoRevillaRe: Show that absolute sum converges
- October 14th 2012, 11:22 AMDinkydoeRe: Show that absolute sum converges
yes sorry, I was editing it :/

thanks for your reply though. It looks so convincing to me...and i need it for something. But i cant show it - October 14th 2012, 11:45 AMVlasevRe: Show that absolute sum converges
There is a special case that is true with . Suppose that and . Then

directly by the Chebyshev inequality, and you are done.

I have a feeling that your statement may be false in general but I'm having trouble finding counter-examples. - October 14th 2012, 11:52 AMDinkydoeRe: Show that absolute sum converges
I am thinking the following: it follows and implying that are Cauchy with limit resp.

Then product as well.... (don't know the proof but product of 2 Cauchy series is something that should be true)

If I'm correct then it should follow that

which is the same as saying that implies

Right? O.o - October 14th 2012, 12:01 PMVlasevRe: Show that absolute sum converges
Actually, here's a proof.

Let and . Then you have that

Next,

Summing over and dividing by yields

The latter two terms converge to 0 by what's given. We use the triangle inequality on the first term

.

This converges to 0 as well, so in the end, the sum on the LHS of the inequality also converges to 0. - October 14th 2012, 12:03 PMDinkydoeRe: Show that absolute sum converges
haha you genious. Thank you kindly :)

- October 14th 2012, 12:06 PMVlasevRe: Show that absolute sum converges
Haha, no problem. It's a cool question though