The change of variable operation is illustrated by the following equation:
$$\sum_{R(i)}a_i = \sum_{R(j)}a_j = \sum_{R(p(i))}a_{p(j)}$$
The equation represents two kinds of transformations:
- Changing the name of the index variable from $i$ to $j$: $\sum_{R(i)}a_i = \sum_{R(j)}a_j$
- Replacing the index by some permutation of its values, represented by $p(j)$: $\sum_{R(i)}a_i = \sum_{R(p(i))}a_{p(j)}$, where, for each integer $i$ satisfying the relation $R(i)$, there must be exactly one integer $j$ satisfying the relation $p(j) = i$.
An example that utilises the first kind of transformation is then provided as follows:
$$\begin{align*} S_1 = \sum_{i=0}^{n}\sum_{j=0}^{i}a_ia_j & = \sum_{j=0}^{n}\sum_{i=j}^{n}a_ia_j &&\text{(change of order)} \\& = \sum_{i=0}^{n}\sum_{j=i}^{n}a_ia_j &&\text{(change of variable)} \end{align*}$$
In the second line, we have interchanged the names of $i$ and $j$ (recognising $a_ja_i = a_ia_j$). If we denote the latter sum by $S_2$, we have
$$2S_1 = S_1 + S_2 = \sum_{i=0}^{n}\left(\sum_{j=0}^{i}a_ia_j + \sum_{j=i}^{n}a_ia_j\right)$$
Question
I'm confused by the stuff in the brackets - why are we allowed to add the change-of-variable sum to the original sum?
If we assume $n=5$, then $\sum_{i=0}^{n}\sum_{j=0}^{i}a_ia_j$ denotes the sum of all the yellow terms below:
But how do we know that $\sum_{i=0}^{n}\sum_{i=j}^{n}a_ia_j$ denotes the sum of the same terms and not these:
Is this implied by simply denoting the second sum by $S_2$ ($S_2$ denoting the same series, but with its order flipped)?
