I want to label each cell in a table and then create the formula of a summation over each column.
I'm new to math and not really sure if my syntax is correct. Espacially the labeling over 3 layers like $c_{x_t}$ and the labeling of the cells like $E_{c_1,t_2}$. Can I label it like that?
Example: $$ \begin{array}{c|lcr} c_x & t_1 & t_2 & t_n \\ \hline c_1 & E_{c_1,t_1} & E_{c_1,t_2} & E_{c_1,t_n} \\ c_2 & E_{c_2,t_1} & E_{c_2,t_2} & E_{c_2,t_n} \\ c_n & E_{c_n,t_1} & E_{c_n,t_2} & E_{c_n,t_n} \\ SUM & S(t_1;c_1;c_n) & S(t_2;c_1;c_n) & S(t_n;c_1;c_n) \end{array} $$
Get SUM from the column $t_1$ over $c_1$ to $c_n$:
$St(t_x;c_{x_f};c_{x_t}) = \sum_{k=c_{x_f}}^{c_{x_t}} E(c_k;t_x)$
Thanks alot!
You notation is unclean in a way: You actually want the sum to be $$S(t_1;c_1;c_n) := \sum_{k=1}^n E_{c_k, t_1}$$
The problem here is how are you going to assure that given the value of $c_1$ and $c_n$, you know from where to where to sum? In other words, if $c_1, \ldots, c_n = 1, \ldots, 1$ then what is $S(t_1;1;1)$?
Is it $\sum_{k=1}^1 E_{c_k,t_1} = E_{c_1, t_1} = E_{1,t_1}$ or is it $\sum_{k=1}^n E_{c_k,t_1} = n E_{1, t_1}$? Or even some other sub-summation?
A clean approach would be instead to tell $S$ about the indices to sum from and to:
$$S(t;i_1;i_2) := \sum_{k=i_1}^{i_2} E_{c_k,t}$$
This prevents any ambiguity in the notation.