# Help with Summations

• Mar 2nd 2013, 07:03 PM
banker
Help with Summations
Hey,

I am having trouble with the following proof.

If Pj = ∑ Qj (limits i = j+1 to ∞), then how do you prove that : ∑ Pj (limits 0 to ∞) = ∑ j*Qj (limits j+1 to ∞)

Cheers.
• Mar 2nd 2013, 10:32 PM
Re: Help with Summations
If
$P_j = \sum\limits_{i=j+1}^\infty Q_i$, then
$\sum\limits_{j=0}^\infty P_j = \sum\limits_{j=0}^\infty (\sum\limits_{i=j+1}^\infty Q_i)= (\sum\limits_{i=1}^\infty Q_i) + (\sum\limits_{i=2}^\infty Q_i) + ... + (\sum\limits_{i=j+1}^\infty Q_i)$

If you can show $(\sum\limits_{i=1}^\infty Q_i) = (\sum\limits_{i=2}^\infty Q_i) = ... = (\sum\limits_{i=j+1}^\infty Q_i)$ then you are done (since there are j terms in the sum). Otherwise I'm stumped.
• Mar 3rd 2013, 08:24 AM
banker
Re: Help with Summations
Cheers man, managed to do it. Same logic but a different intermediary step!
• Mar 3rd 2013, 11:23 AM
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