How Can Adding Forever Give a Finite Answer? Each Step Smaller Than the Last

Each term is obtained by multiplying the previous one by the same fixed number, so the sequence is determined entirely by where it starts and what that multiplier is. Multiplying by two gives one, two, four, eight and so on, growing without limit. Multiplying by a half gives one, a half, a quarter, an eighth, shrinking towards zero. The behaviour of the sum depends entirely on whether the multiplier is greater or less than one in size. If it is greater, the terms grow and the total grows without limit. If it is less, the terms shrink fast enough that the total approaches a definite value however many are added.

The reason is visible in a simple picture:

• •Take a square of area one and shade half of it
• •Shade half of what remains, which is a quarter
• •Shade half of what remains again, which is an eighth
• •Continuing forever shades the whole square and never more
• •So a half plus a quarter plus an eighth and onwards totals exactly one
• •The same reasoning works for any multiplier smaller than one

Zeno of Elea argued in the fifth century before the common era that motion is impossible, because reaching a destination requires first covering half the distance, then half of what remains, then half of that, through infinitely many stages, which appears to require infinite time. The arithmetic above answers the version about distance directly, since the infinitely many pieces sum to the finite whole. Answering the version about time requires noticing that the time taken for each stage also halves, so those intervals form the same kind of shrinking sum and total a finite time as well. Whether that fully disposes of the philosophical argument is debated, and it certainly disposes of the mathematical difficulty.

The intuition that infinitely many shrinking terms must total something finite is wrong, and the counterexample is worth knowing because it is the standard warning in the subject. Adding one, then a half, then a third, then a quarter and so on forever produces a total that grows without limit, even though the terms shrink towards zero. The growth is extraordinarily slow, needing enormous numbers of terms to reach a modest total, which is why the result surprises people. The demonstration groups the terms in blocks whose sums each exceed a half, and since there are infinitely many such blocks the total exceeds any bound. Terms shrinking is therefore necessary for a finite total and nowhere near sufficient.

The pattern appears throughout practical mathematics. Compound interest is exactly this, since each period multiplies the balance by a fixed factor, and the formulas for repayments on a loan and for the value of an annuity are sums of such a series. Radioactive decay halves a quantity in each fixed period. The total value of a perpetual income stream discounted at a constant rate is one of these sums, which is the basis of valuing assets. Repeating decimals are these sums, which is why a third written as a decimal repeats forever and equals exactly a third. Fractal constructions add shrinking pieces at each stage, and whether the total area or length is finite depends on exactly this test.

Each term is a fixed multiple of the last, and the total is finite whenever that multiple is smaller than one, because the terms shrink fast enough. Shading half a square, then half the remainder, then half again shows why the pieces total exactly one. That arithmetic answers Zeno's argument about covering infinitely many halves, since the times also halve. Compound interest, radioactive decay and repeating decimals all follow the pattern.