Forever Sums

Cut a cake in half, then cut the rest in half, and keep going. The pieces are 1/2, 1/4, 1/8, ... of the cake, and they can never add up to more than the whole cake. A sum that goes on forever can have a finite answer.

Look at the sum formula $S = \frac{a\left(1\right)\cdot \left(1-{r}^{n}\right)}{1-r}$. When r is between -1 and 1, ${r}^{n}$ shrinks towards 0 as n grows: ${\frac{1}{2}}^{10}$ is already less than a thousandth. So for a sum that never ends, the ${r}^{n}$ disappears and

$$S = \frac{a\left(1\right)}{1-r}$$

For $10+5+\frac{5}{2}+...$, the ratio is 1/2, so $S = \frac{10}{1-\frac{1}{2}}$, which is 20. If r is 1 or more, or -1 or less, the terms never shrink, and the sum has no finite value.