Confusion in notation of the finite double series $\sum_{j=1}^{m} \sum_{i=1}^{n} i\cdot j^2$

Question

Evaluate: $$\sum_{j=1}^{m} \sum_{i=1}^{n} i\cdot j^2$$

• Now I am confused how to evaluate the sum as I think that the question hasn't specified that whether $m>n$ or $n>m$.
• I don't know where the series would terminate.

Could anyone clarify the notation as well as give an expanded version of the above expression?

Answer 1

$$ \sum_{j=1}^{m} \sum_{i=1}^{n} ij^2 =\sum_{j=1}^{m} \left(\sum_{i=1}^{n} ij^2\right) =\sum_{j=1}^{m} \left(j^2\sum_{i=1}^{n} i\right) =\sum_{j=1}^{m} \left(j^2\left(\sum_{i=1}^{n} i\right)\right) =\left(\sum_{j=1}^{m} j^2\right)\left(\sum_{i=1}^{n} i\right) $$

Answer 2

$$\sum\limits_{j=1}^m\sum\limits_{i=1}^n ij^2=\sum\limits_{j=1}^m j^2\sum\limits_{i=1}^n i$$$$=\sum\limits_{i=1}^n j^2\cdot\big(\frac{n\cdot(n+1)}2\big)$$$$=\bigg(\frac{m\cdot(m+1)\cdot(2m+1)}6\bigg)\cdot\bigg(\frac{n\cdot(n+1)}2\bigg)$$