# Sums with boundaries of sums

Lately I came upon sums with boundaries other sums:
$\sum_{i=1}^{\sum_{j=1}^{X}p_j}q_i$
Where X can also be a sum (with a boundary of sum) etc.
I wanted to ask what kind of mathematics should I use to simplify or transform these sums in other formats (if possible) because I have no idea how to deal with them.

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$\sum_{j=1}^{X}p_j$ is just a number (or perhaps a random variable if $X$ also is), so one approach would be to calculate it first. – Henry Aug 6 '12 at 14:42
I cannot do that because in the general case I theoretically can have an infinite tower of sums... Assuming that the boundary has a value (which will eventually have) is just hiding the complexity behind a symbol. – Evangelos Bempelis Aug 6 '12 at 14:47

If you have an infinite tower, the only way I can see to approach it is to see it as a sequence. The first member is $a_1=\sum_{i=1}^1q_i=q_1$. The second is $a_2=\sum_{i=1}^{\sum_{j=1}^{X}p_j}q_i$. Then $a_3=\sum_{i=1}^{\sum_{j=1}^{{\sum_{k=1}^{Y}r_k}}p_j}q_i$. If you can prove that the sequence converges, you are home. Sometimes you can get there by assuming that one more step doesn't change anything, so the limit $L$ satisfies $L=\sum L$ with appropriate limits on the sum.