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Q: infinte sereis ( Answered,   0 Comments )
Question  
Subject: infinte sereis
Category: Miscellaneous
Asked by: fmunshi-ga
List Price: $5.00
Posted: 20 Jul 2003 20:50 PDT
Expires: 19 Aug 2003 20:50 PDT
Question ID: 233190
suppose teh series sum(ak) converges absoultly and taht bk is bounded
.  Show that sum(ak*bk) converges absolutely
Answer  
Subject: Re: infinte sereis
Answered By: elmarto-ga on 20 Jul 2003 23:01 PDT
 
Hello again fmunshi!
Since bk is bounded, we know that there exists some bound M such that

|bk| < M for all k

We also know that sum(|ak|)=N, (where N, of course, is finite),
because sum(ak) is absolutely convergent.

We have to show that sum( |ak*bk| ) = L (where L, of course is a
finite number). This means that the series sum(ak*bk) converges
absolutely. Since M is a bound for we bk, we have

  |ak*bk|
= |ak|*|bk|
<=|ak|*M

Therefore,

  sum(|ak*bk|)
<=sum(M*|ak|)
 =M*sum(|ak|)
 =M*N

So, sum(|ak*bk|) is less than or equal to M*N (which is a finite
number) and greater than 0 (because |ak*bk| is greater than 0). This
implies that the series sum(|ak*bk|) converges to some number between
0 and M*N.


I hope this was clear enough. If you have any doubts regarding this
answer, please don't hesitate to request a clarification. Otherwise, I
await your rating and final comments.


Best wishes!
elmarto
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