Q.Find the sum to infinity of the Geometric Progression $8 + 4 + 2 + 1 + \cdots$

Check $|r| < 1$, then use $S_\infty = \frac{a}{1 - r}$.

Here $a = 8$ and $r = \dfrac{4}{8} = \dfrac12$ (checked: $\dfrac{2}{4} = \dfrac12$, $\dfrac{1}{2} = \dfrac12$). Since $|r| = \tfrac12 < 1$, the sum to infinity exists and equals

$S_\infty = \frac{a}{1 - r} = \frac{8}{1 - \tfrac12} = \frac{8}{\tfrac12} = 16.$ …

Applying $S_\infty = \frac{a}{1-r}$ when $|r| \ge 1$ — the formula is valid only for $|r| < 1$. For a GP such as $2 + 4 + 8 + \cdots$ ($r = 2$) the terms grow, the total is unbo …