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.

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    $\begingroup$ So you mean $\sum_{j=1}^m \sum_{i=1}^k n^{(j)}_i$ where the upper index indicates the different summation values? Or written in your way: $ \sum_{i=1}^k n^{(1)}_i + \sum_{i=1}^k n^{(2)}_i + \ldots + \sum_{i=1}^k n^{(m)}_i$? $\endgroup$ – ctst Nov 18 '16 at 12:27
  • $\begingroup$ i don't think so, see every series has different values, as example i need to calculate the following { (3) + ( 4+2+6) + (7+1+3))} m=3 k= 1,3,3 $\endgroup$ – aram Nov 18 '16 at 12:36

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*}


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