Infinite Geometric Progression

An infinite geometric progression is a GP whose terms go on forever — there is no last term. The key question is whether such a sum can ever be a finite number. It can, but only when the common ratio $r$ lies strictly between $-1$ and $1$; in that case the terms shrink fast enough that the total converges to $\frac{a}{1-r}$. This simple formula is one of the most powerful tools in mathematics, l …