$$ \sum_{i=1}^\infty \sum_{k=1}^i p_X(i) = \sum_{i=1}^\infty i p_X(i). $$ This is from a problem solution in introductory probability theory course (problem 3a here, solution here) . X is a random variable that takes nonnegative integer values. The expression implies that sum from 1 to i of probabilities of X being i is equal to i (inner sum collapses into i). But I'm not sure why that is the case.
I understand that 1) probabilities for ALL i's must add up to 1, and 2) if we take 1 i times (the summation) we should get i. But it doesn't connect... Perhaps not fully getting notation here?
Any help would be much appreciated.
This is not about probabilities adding up to $1.$ Suppose, for example, that $i=4.$ Then $$ \sum_{k=1}^i p_X(i) = \sum_{k=1}^4 p_X(4) = \underset{\Large\underset{k=1}\uparrow}{p_X(4)} + \underset{\Large\underset{k=2}\uparrow}{p_X(4)} + \underset{\Large\underset{k=3}\uparrow}{p_X(4)} + \underset{\Large\underset{k=4}\uparrow}{p_X(4)} = 4p_X(4) = ip_X(i). $$
Note that in the expression $p_X(i),$ the index $k$ that goes from $1$ to $4$ does not appear. That is why all four terms are the same as each other.