Sum of Geometric Sequence/Examples/Index to Minus 2

Theorem

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

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

breaks down when $n = -2$:

$\ds \sum_{j \mathop = 0}^{-2} x^j \ne \frac {x^{-1} - 1} {x - 1}$

$\ds \sum_{j \mathop = 0}^{-2} x^j = 0$

while on the right hand side we have:

$\ds \frac {x^{\paren {-2} + 1} - 1} {x - 1}$

$=$

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

$\ds $

$=$

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

$\ds $

$=$

$\ds \frac {\paren {1 - x} / x} {x - 1}$

$\ds $

$=$

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

$\ds $

$=$

$\ds -\frac 1 x$

$\blacksquare$