what does $\sum\limits_{i=1}^{10} \sum\limits_{j=1}^{10} \min(i,j)$ mean

Question

what does

$$\sum\limits_{i=1}^{10} \sum\limits_{j=1}^{10} \min(i,j)$$

mean?

I'm aware that $\sum$ represents the summation, and min the minimum value . But when written like $\sum \sum$ , what exactly does it mean?

Answer 1

This is called a double summation. Since the summations are all finite, you can read it as either $$\sum_{i=1}^{10} \left( \sum_{j=1}^{10} \left( \min(i,j) \right) \right)$$ or $$\sum_{j=1}^{10} \left( \sum_{i=1}^{10} \left( \min(i,j) \right) \right)$$ or even as (if you prefer) $$\sum_{i,j=0}^{10}\min(i,j)$$ which some people are more accustomed to.

Basically, it is the sum of $\min(i,j)$ for $i,j$ ranging over $[0,10]$.

Answer 2

Hint

Evaluating the expression $\ \sum_\limits{i=1}^{10}\sum_\limits{j=1}^{10}\min(i,j)\ $ is facilitated by dividing the inner sum into two parts: \begin{align} \sum_\limits{j=1}^{10}\min(i,j)&=\sum_{j=1}^ij+\sum_{j=i+1}^{10}i\\ &=\sum_{j=1}^ij+(10-i)i\\ &=\frac{i(i+1)}{2}+(10-i)i\ \end{align} for $\ 1\le i\le9\ $, because $\ \min(i,j)=j\ $ when $\ j\le i\ $ and $\ \min(i,j)=i\ $ when $\ j>i\ $, and \begin{align} \sum_\limits{j=1}^{10}\min(10,j)&=\sum_\limits{j=1}^{10}j\\ &=45 \end{align} because $\ \min(10,j)=j\ $ whenever $\ 1\le j\le10\ $.