What is the correct way to write a sum? I want to write a general formulation for the voltage at the last node of a line of an electrical grid. Here is an example line:
I have written:
\begin{equation} u_\mathrm{last}=u_0-\sum_{l\in\mathcal{L}}\Big(Z_l\cdot\sum_{\substack{n\in\mathcal{N}\\n\geq l}}I_n\Big) \end{equation}
where $\mathcal{L}=\{1, 2, 3, 4\}$ is the set of lines and $\mathcal{N}=\{1, 2, 3 , 4\}$ is the set of nodes. $Z_l$ is the impedance of the $l$-th line and $I_n$ is the current into the $n$-th node (attached loads, that are not depicted in the picture). $u_0$ is the voltage at the transformer low voltage side. So for example the last voltage node of a line with 3 nodes would be:
$$ u_3=u_0-\big(Z_1\cdot(I_1+I_2+I_3)+Z_2\cdot(I_2+I_3)+Z_3\cdot I_3 \big) $$
Is my general formulation above correct? Because my Professor (he is not a mathematician, just like me) thinks, that sum operators allways need upper and lower bounds, like:
$$ \sum_{l_0}^{l_n} $$
But with these upper and lower bounds, Im not sure how to properly express, that the sum of $I$ is getting one element shorter each iteration of the outer sum.
Thanks for any help in advance!
There are usually more than one way to represent a sum and your approach is fine. In fact considering $3$ nodes with index sets \begin{align*} \mathcal{N}=\{1,2,3\}=\mathcal{L} \end{align*} together with a special node $u_0$ we have the equality chain \begin{align*} \color{blue}{u_3}&=u_0-\left(Z_1\cdot(I_1+I_2+I_3)+Z_2\cdot(I_2+I_3)+Z_3\cdot I_3 \right)\\ &\color{blue}{=u_0-\sum_{l\in\mathcal{L}}Z_l \sum_{\substack{n\in\mathcal{N}\\n\geq l}}I_n} \color{blue}{=u_0-\sum_{l=1}^3Z_l\sum_{n=l}^3 I_n}\tag{1} \end{align*}
As we can see at the left-hand side of (1), lower and upper bound need not be explicitly stated. They are nevertheless implicitly given by definition of $\mathcal{N},\mathcal{L}$ and the constraint $n\geq l$.
Another aspect is, that $l\in\mathcal{L}$ does not imply any order of summation. We just have to take each element from $\mathcal{L}$ in any order we like. This is admissible since addition is commutative.
Hint: You might find chapter 2: Sums in Concrete Mathematics by R.L. Graham, D.E. Knuth and O. Patashnik helpful. It provides a thorough introduction in the usage of sums.