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 = a(1) × (1 - r^n)/(1 - r). When r is between -1 and 1, r^n shrinks towards 0 as n grows: (1/2)^10 is already less than a thousandth. So for a sum that never ends, the r^n disappears and

S = a(1)/(1 - r)

For 10 + 5 + 5/2 + ..., the ratio is 1/2, so S = 10/(1 - 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.