Suppose I have $$ \sum_{n=1}^{m}n \ \ + \ \ \sum_{k=1}^{m}k.$$ This isn't a generalized case, but I'm not sure what the extent of the generalized case is to begin with. Can I assume that $n=k$ and have both summations count to $m$ at simultaneously?
What if I had $$ \sum_{n=1}^{m}n \ \ * \ \ \sum_{k=1}^{m}k.$$
or any operation performed on multiple summations?
This follows the pattern of adding similar powers of an expansion. For instance consider \begin{align} S(t) &= \sum_{n=0}^{m} \frac{a_{n-m}}{t^{m-n}} + \sum_{k=1}^{m} a_{k} \, t^{k} \\ &= \frac{a_{-m}}{t^{m}} + \frac{a_{-m+1}}{t^{m-1}} + \cdots + \frac{a_{-1}}{t} + a_{0} + a_{1} \, t + \cdots + a_{m} \, t^{m} \\ &= \sum_{k=-m}^{m} a_{k} \, t^{k}. \end{align}
For the first proposed example of the problem it is seen that $$\sum_{n=1}^{m} n + \sum_{k=1}^{m} k = 2 \, \sum_{k=1}^{m} k = 2 \, \binom{m+1}{2}.$$
A similar prospect can be made for multiplying series. In the case provided then $$\left(\sum_{n=1}^{m} n \right)\left(\sum_{k=1}^{m} k \right) = \left(\sum_{n=1}^{m} n \right)^2 = \binom{m+1}{2}^2.$$
