1. ## 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.

2. ## 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.

3. ## Re: Help with Summations

Cheers man, managed to do it. Same logic but a different intermediary step!

4. ## Re: Help with Summations

I'm curious to what the different intermediary step is.