Summation of a real sequence

• February 13th 2007, 10:29 AM
led5v
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.
• February 13th 2007, 11:17 AM
ThePerfectHacker
Quote:

Originally Posted by led5v
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.

..
• February 13th 2007, 11:29 AM
led5v
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.
• February 14th 2007, 08:52 AM
led5v
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,