Sum of Geometric Sequence/Examples/Index to Minus 1

Theorem

Let $x$ be an element of one of the standard number fields: $\Q, \R, \C$ such that $x \ne 1$.

$\ds \sum_{j \mathop = 0}^n x^j = \frac {x^{n + 1} - 1} {x - 1}$

still holds when $n = -1$:

$\ds \sum_{j \mathop = 0}^{-1} x^j = \frac {x^0 - 1} {x - 1}$

$\ds \sum_{j \mathop = 0}^{-1} x^j = 0$

while on the right hand side we have:

$\ds \frac {x^{\paren {-1} + 1} - 1} {x - 1}$

$=$

$\ds \frac {x^0 - 1} {x - 1}$

$\ds $

$=$

$\ds \frac 0 {x - 1}$

$\ds $

$=$

$\ds 0$

as long as $x \ne 1$.

However, the theorem itself is based on the assumption that $n \ge 0$, so while the result is correct, the derivation to achieve it is not.

$\blacksquare$