Not sure how to interpret nested sums with numbers in between.

Question

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I'm not looking for help with this specific question. I'm just not sure how to deal with nested sums with the numbers between the sigmas. For example, on the first iteration the innermost sum is -16, should that number be multiplied by m, the the middle sum be taken?

Answer 1

Let's use a simpler example:

$$\sum_{k=1}^2k\sum_{m=0}^k m\sum_{n=-1}^m \sin(n+m)$$

This breaks down as follows:

$$\begin{align*} & 1\cdot 0\cdot \big(\sin(-1+0)+\sin(0+0)\big) \\ + & 1\cdot 1\cdot \big(\sin(-1+1)+\sin(0+1)+\sin(1+1)\big) \\ + & 2\cdot 0\cdot \big(\sin(-1+0)+\sin(0+0)\big) \\ + & 2\cdot 1\cdot \big(\sin(-1+1)+\sin(0+1)+\sin(1+1)\big) \\ + & 2\cdot 2\cdot \big(\sin(-1+2)+\sin(0+2)+\sin(1+2)+\sin(2+2)\big)\end{align*}$$