I know of the arithmetic series but I do not think this can be applied here. I have the expression $((j \cdot k)^2k + (j \cdot k)^2j)$. Is there any way to simplify/compute $$ \sum_{j_1 = 1}^{j}{\sum_{k_1 = 1}^{k}{((j_1 \cdot k_1)^2k_1 + (j_1 \cdot k_1)^2j_1)}} $$ That is, for all $j \cdot k$ pairs.
Thank you.
There's a typo in your "Big expression"
$$\Sigma_{j_1=1}^j \Sigma_{k_1=1}^k j_1^3k_1^2+j_1^2k_1^3$$
$$=(\Sigma_{j_1=1}^j j_1^3)(\Sigma_{k_1=1}^k k_1^2)+(\Sigma_{j_1=1}^j j_1^2)(\Sigma_{k_1=1}^k k_1^3)$$
Now evaluate each part separately, since they're independent (I assume you know $\Sigma x^2$ and $\Sigma x^3$)
This is because when you're evaluating $\Sigma_x \Sigma_y f(x)g(y)$ you can treat the $f(x)$ inside the $\Sigma_y$ as constant and take it out.
Cheers :)