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.
If
$\displaystyle P_j = \sum\limits_{i=j+1}^\infty Q_i$, then
$\displaystyle \sum\limits_{j=0}^\infty P_j = \sum\limits_{j=0}^\infty (\sum\limits_{i=j+1}^\infty Q_i)= (\sum\limits_{i=1}^\infty Q_i) + (\sum\limits_{i=2}^\infty Q_i) + ... + (\sum\limits_{i=j+1}^\infty Q_i)$
If you can show $\displaystyle (\sum\limits_{i=1}^\infty Q_i) = (\sum\limits_{i=2}^\infty Q_i) = ... = (\sum\limits_{i=j+1}^\infty Q_i)$ then you are done (since there are j terms in the sum). Otherwise I'm stumped.