General Distributivity Theorem/Warning

Warning about use of of General Distributivity Theorem

When using the General Distributivity Theorem, be careful not to conflate the indices.

The following is not a valid argument:

$\ds \left({\sum_{i \mathop = 1}^n a_i}\right) \left({\sum_{j \mathop = 1}^n \frac 1 {a_j} }\right)$

$=$

$\ds \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n \frac {a_i} {a_j}$

$\ds $

$=$

$\ds \sum_{i \mathop = 1}^n \sum_{i \mathop = 1}^n \frac {a_i} {a_i}$

$\ds $

$=$

$\ds \sum_{i \mathop = 1}^n 1$

simplification

$\ds $

$=$

$\ds n$

The second step uses the same index $i$ for two different purposes.

Thus elements of the form $\dfrac {a_i} {a_j}$ where $i \ne j$ are not represented in the summation.

$\blacksquare$

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 $25$