Jorge Valle

Performing double summations

The double summation notation is just slightly more challenging than the summation notation, specially when we break down the expression into an inner and outer sum.

At the end of the day, we will still end up with just one number: the result of the double sum. Let's take the following example.

$$ \sum_{i=1}^2 \sum_{j=1}^3 (i + j) $$
Figure 1: A double summation.

Notice that we have two indices; $i$ and $j$, but we only have one operation we are performing on them together.

The inner sum

Let's first identify the inner sum, enclosed in the square brackets.

$$ \sum_{i=1}^2 \left[ \sum_{j=1}^3 (i + j) \right] $$
Figure 2: The inner sum.

The index starts at one, and ends at three. Taken by itself, and just carrying the variable of the outer sum, this inner sum could be expressed more simply as follows.

$$ \begin{align} \sum_{j=1}^3 (i + j) & = i + 1 + i + 2 + i + 3 \\ & = 3i + 6 \end{align} $$
Figure 3: Performing just the inner sum.

The outer sum

Now that we've simplified what the inner sum looks like, which is $3i + 6$, we can just plug in that expression and run through the indices of the outer sum. Here we're just going from one to two.

$$ \begin{align} \sum_{i=1}^2 (3i + 6) & = (3 \cdot 1 + 6) + (3 \cdot 2 + 6) \\ & = (3 + 6) + (6 + 6) \\ & = 9 + 12 \\ & = 21 \end{align} $$
Figure 4: Plugging in inner sum into double sum.

The result of the double summation is $21$. This is the simplest method I know of to conceptualize of these double summations. It can be checked on the following online double summation calculator.