Q.Find the sum to infinity of the G.P. $5, \dfrac53, \dfrac59, \ldots$

A Geometric Progression can be summed all the way to infinity only when $|r|<1$, since only then does $r^n\to0$ as $n$ grows without bound. Taking the limit of the finite-sum formula $S_n=\dfrac{a(1-r^n)}{1-r}$ under this condition gives $S_\infty=\dfrac{a}{1-r}$ — a single finite number, even though infinitely many terms are being added. If $|r|\ge1$, no such finite sum exists and the formula must never be applied. This idea gives a clean method for converting any purely recurring decimal to a fraction (e.g. $0.\overline{d}=\dfrac{d}{9}$), by recognising the decimal expansion as an infinite G.P. with ratio $\dfrac1{10}$.