TAOCP 1.2.3 Exercise 4
For $n = 3$, the left-hand side of Eq.
Section 1.2.3: Sums and Products
Exercise 4. [10] Without using the $\sum$-notation, write out the equivalent of each side of Eq. (10) as a sum of sums for the case $n = 3$.
Verified: yes
Solve time: 45s
For $n = 3$, the left-hand side of Eq. (10) is
$$ \sum_{i=1}^3 \sum_{j=1}^i a_{ij} = (a_{11}) + (a_{21} + a_{22}) + (a_{31} + a_{32} + a_{33}). $$
The right-hand side of Eq. (10) is
$$ \sum_{j=1}^3 \sum_{i=j}^3 a_{ij} = (a_{11} + a_{21} + a_{31}) + (a_{22} + a_{32}) + (a_{33}). $$
Both expressions contain exactly the same terms, namely all $a_{ij}$ such that $1 \le j \le i \le 3$, grouped in different ways.