Summation of i from 1 to n of Summation of j from 1 to i/Example
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Example of Summation of $i$ from $1$ to $n$ of Summation of $j$ from $1$ to $i$
- $\ds \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^i a_{i j} = \sum_{j \mathop = 1}^n \sum_{i \mathop = j}^n a_{i j}$
Let $n = 3$.
\(\ds \sum_{i \mathop = 1}^3 \sum_{j \mathop = 1}^i a_{i j}\) | \(=\) | \(\ds \sum_{j \mathop = 1}^1 a_{1 j} + \sum_{j \mathop = 1}^2 a_{2 j} + \sum_{j \mathop = 1}^3 a_{3 j}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \left({a_{1 1} }\right) + \left({a_{2 1} + a_{2 2} }\right) + \left({a_{3 1} + a_{3 2} + a_{3 3} }\right)\) |
\(\ds \sum_{j \mathop = 1}^3 \sum_{i \mathop = j}^3 a_{i j}\) | \(=\) | \(\ds \sum_{i \mathop = 1}^3 a_{i 1} + \sum_{i \mathop = 2}^3 a_{i 2} + \sum_{i \mathop = 3}^3 a_{i 3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \left({a_{1 1} + a_{2 1} + a_{3 1} }\right) + \left({a_{2 2} + a_{3 2} }\right) + \left({a_{3 3} }\right)\) |
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products: Exercise $4$