Limit of double series

elimqiu · 发表于 2010-9-29 04:17

Yes I can see the proof suggested in the book is not ';neat';. Let me see what it should be. I haven';t question myself about those questions for a long time...

elimqiu · 发表于 2010-9-29 04:46

Why we can take the limit with respect to n? Since it a finite sum with respect to m.

kindlychung · 发表于 2010-9-29 12:08

Nice. My mind is much clearer now.
The key lies in the switching of m and n.

elimqiu · 发表于 2010-9-29 12:15

Glad to know that you got it so quickly.
I didn';t mention but actually I proved that Sn,n → s in the middle of my proof.

kindlychung · 发表于 2010-9-29 12:48

elimqiu · 发表于 2010-9-29 13:00

Let n=m in the 6th line of my proof, you get Sn,n

kindlychung · 发表于 2010-9-29 13:08

[这个贴子最后由kindlychung在 2010/09/29 01:20pm 第 2 次编辑]

elimqiu · 发表于 2010-9-29 13:40

[这个贴子最后由elimqiu在 2010/09/29 06:57am 第 1 次编辑]

Let Q(n) be the sum of |a(i,j)| with  1≤i,j≤n
Try to show that  {Q(n)}  is a Cauchy sequence and so        
2m
|∑ d(k) - Sm,m | ≤ Q(2m)-Q(m) < ε   when m is big enough
2

kindlychung · 发表于 2010-9-29 18:01

Very well illustrated, I can see you point.
But...

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Limit of double series

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