Summation/Inequality/Examples/From 1 to Pi

Example of Summation by Inequality

Let $n = 3 \cdotp 1 4$.

Then:

$\ds \sum_{1 \mathop \le j \mathop \le n} a_j = a_1 + a_2 + a_3$

Proof

$\ds \sum_{1 \mathop \le j \mathop \le n} a_j$

means:

The sum of all $a_j$ where $j \in \Z$ and $1 \le j \le n$

from which it follows that $j \in \set {1, 2, 3}$.

Hence the result.

$\blacksquare$

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 $1$