Infinite Geometric Series

Adding infinitely many positive numbers and getting a finite answer sounds like it should be impossible, so here is the argument in one line. Everything after the first term is the whole series again, shrunk by the ratio: $S = a_1 + rS$. Rearranged, $S(1-r) = a_1$, so $S = a_1/(1-r)$. That self-similarity is the real content — the tail beyond any point is a scaled copy of the entire thing — and it also shows why $|r| < 1$ is not a technicality bolted on afterwards. If the copy is not smaller than the original, the argument has nothing to stand on.

Zeno's runner is the classic instance: covering half the remaining distance forever gives $\tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8} + \dots = (0.5)/(1 - 0.5) = 1$, the whole journey, and the paradox dissolves. Repeating decimals are the same collapse: $0.7777\ldots$ is $0.7 + 0.07 + 0.007 + \dots$ with $a_1 = 0.7$ and $r = 0.1$, so $S = 0.7/0.9 = 7/9$, and this is the honest proof that $0.999\ldots = 1$ rather than a trick.

The applied version shows up wherever something is added repeatedly while a fraction of what is already there persists. If a dose is repeated at fixed intervals and 40% of each dose is still present when the next arrives, the level climbs toward a steady state of $1/(1-0.4) = 1.67$ doses, not toward infinity — which is why repeat dosing plateaus, whether the thing being dosed is a medication or a chemical in a recirculating loop. The plateau is reached in practice long before "forever": the sum is within 1% of its limit after about $\ln(0.01)/\ln(r)$ terms, which at $r = 0.4$ is five.

Three errors. The first and commonest is misidentifying $a_1$: it is the first term you are actually adding, not the quantity that generated it. In $0.7 + 0.07 + \dots$, $a_1$ is 0.7, not 7 and not 0.777. If a series begins partway along, that term is your $a_1$. The second is assuming the ratio is constant without checking — compute $a_2/a_1$ and $a_3/a_2$ and confirm they agree before trusting any of this, because a sequence that merely decreases is not necessarily geometric. The third is feeding in $|r| \ge 1$. At $r = 1$ every term is identical and the total runs away; at $r = -1$ the partial sums flip between 1 and 0 forever and never settle on anything. Beyond that the formula will still return a number — put $r = 2$ and $a_1 = 1$ and it says $1 + 2 + 4 + 8 + \dots = -1$ — and that number is not a sum in any ordinary sense. It is worth knowing that such values do mean something under a broader definition of summation, but never as an answer to "how much is this pile."