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I have the following nested double sum :

$\sum_{t=1}^{T}\sum_{u=t}^{T} Z(u) \cdot (1+i)^{-u}$ with $0 and $Z(u)$ being a non-specified function.

By working with an example of $T=3$ I suppose the solution should be

$\sum_{t=1}^T t \cdot Z(t) \cdot (1+i)^{-t}$.

Is that correct?

My problem is to figure out which calculation rule for sums is to apply here when you have the index of the outer sum as starting point for the second sum. The normal commutative and distributive rule don't seem to apply here. I can hardly argue "It works for $T=3$ so it must be right for all $T$." Could someone please tell me the general/universal calculation rule?

Does/how does the solution change when the sum is changed to

$\sum_{t=1}^{T}\sum_{u=t}^{T} Z(u) \cdot (1+i)^{-u+t}$ ?

3 Answers 3