[Math] the summation of multiple sums

Question

I have a problem where I need to formalize the following summation

$\sum_{i=1}^k ni $ + $\sum_{i=1}^k ni $ + $\sum_{i=1}^k ni$ + …
the sum will be repeted m times and the k values will be different in each summation notation.

Answer 1

Since there are $m$ sums $S_i,1\leq i\leq m$, we can write \begin{align*} \sum_{i=1}^mS_i\qquad\qquad\qquad &m\geq 1 \end{align*}

Each sum $S_i$ contains $k_i$ summands $n_{i,j},1\leq j\leq k_i$ \begin{align*} S_i=\sum_{j=1}^{k_i} n_{i,j}\qquad\qquad\quad 1\leq i\leq m \end{align*}

We obtain \begin{align*} \sum_{i=1}^mS_i=\sum_{i=1}^m\sum_{j=1}^{k_i}n_{i,j}\qquad\qquad m\geq 1 \end{align*}