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Math Help - Summation of a real sequence

  1. #1
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    Summation of a real sequence

    While trying to prove a mathematical relationship, I ended up with the following term

     \sum_{k=0}^{\infty} \sum_{n=0}^k kF(n)G(k-n)

    Where, F and G represent functions that accept inter arguments and return real numbers.

    Now, the final goal of my proof is equal to

     \sum_{x=0}^{\infty} \sum_{y=0}^{\infty}  (x+y)F(x)G(y)

    Can some one please help me if the above two can be proved to be equivalent. I would really appreciate if you could please illustrate all the intermediate steps in this proof.
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  2. #2
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    Quote Originally Posted by led5v View Post
    While trying to prove a mathematical relationship, I ended up with the following term

     \sum_{k=0}^{\infty} \sum_{n=0}^k kF(n)G(k-n)

    Where, F and G represent functions that accept inter arguments and return real numbers.

    Now, the final goal of my proof is equal to

     \sum_{x=0}^{\infty} \sum_{y=0}^{\infty}  (x+y)F(x)G(y)

    Can some one please help me if the above two can be proved to be equivalent. I would really appreciate if you could please illustrate all the intermediate steps in this proof.
    ..
    Attached Thumbnails Attached Thumbnails Summation of a real sequence-picture11.gif  
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  3. #3
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    Thanks for your illustrative reply. Now I understand that both of these terms are equal. But still I can't find a formal argument to support this claim.

    Can this be proved in a formal manner as well using the rules of summations. I would be really thankful if someone could please point me to a formal proof of this goal.
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  4. #4
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    It seems that the senior members in this forum are thinking that I have got my answer, which is not the case. I am still looking for a formal argument to prove the above clain and have looked into a lot of texts on summations but in vain.

    Any help in this regard would be greatly appreciated.
    thanks,
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