General Distributivity Theorem/Warning

Warning about use of of General Distributivity Theorem

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.

$\blacksquare$