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Summation by Parts

Date: 01/07/2004 at 13:36:04
From: Doruk
Subject: Series Problem

Using 'E' to represent sigma, is there an approximate solution to
E(Ai*Bi) = ? where i = 0,1,...,n if Ai is known explicitly and E(Bi)
is known?



Date: 01/07/2004 at 15:12:33
From: Doctor Vogler
Subject: Re: Series Problem

Hi Durok,

I think you are looking for a technique called "summation by parts" 
(analogous to the calculus technique "integration by parts"). 
Essentially, in integration by parts, you change

  int(u*dv) = u*v - int(v*du).

In summation by parts, you change

  sum(u*dv) = u*v - sum(v*du),

where u is the Ai, and dv is the Bi (so that v = sum(Bi)).

More explicitly, it works like this:

  Suppose that S(k) = sum(B(i),i=1 to k), so that S(0) = 0.

  sum(A(i)*B(i), i=1 to n) = sum(A(i)*(S(i)-S(i-1)),i=1 to n)
     = sum(A(i)*S(i),i=1 to n) - sum(A(i)*S(i-1),i=1 to n)
     = sum(A(i)*S(i),i=1 to n) - sum(A(i+1)*S(i),i=0 to n-1)
     = sum(A(i)*S(i),i=1 to n) - sum(A(i+1)*S(i),i=1 to n-1)
     = A(n+1)*S(n) + sum((A(i)-A(i+1))*S(i),i=1 to n)
     = A(n+1)*S(n) - sum((A(i+1)-A(i))*S(i),i=1 to n)

Does that make sense?

See also references on the Internet (found by searching Google for 
"summation by parts", with the quotes):

  Summation By Parts
    http://mathworld.wolfram.com/SummationbyParts.html 

  Lemma:  Summation By Parts
    http://www.shu.edu/projects/reals/numser/proofs/sumparts.html 

See if you can take it from there and write back if you need more help.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 01/07/2004 at 17:39:04
From: Doruk
Subject: Thank you (Series Problem)

Thanks for the quick reply, it was very useful information.  I'm
working on it and now I have another question.  If I want to use only
S(n), not S(i) i=1 to n, what would be the best approximation to the
given exact solution?



Date: 01/07/2004 at 20:09:53
From: Doctor Vogler
Subject: Re: Series Problem

Hi Doruk,

Knowing only the A's and the sum of all of the B's is not enough
information.  The sum of the products could differ wildly according to
the individual values of the B's.  So you need more information to get
any kind of approximation.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 01/08/2004 at 00:14:55
From: Doruk
Subject: Thank you (Series Problem)

Thanks again for all the useful information and help.
Associated Topics:
High School Discrete Mathematics
High School Sequences, Series

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