Halfway to the wall

After $n$ steps you have walked

$$\frac12 + \frac14 + \dots + \frac1{2^n} = 1 - \frac{1}{2^n}$$

of the way, so you are $\tfrac{1}{2^n}$ from the wall. At no finite stage have you arrived. But the gap becomes smaller than any distance you care to name: smaller than a millimetre after 10 steps of a metre, smaller than the width of an atom after about 34. Mathematicians define the sum of infinitely many numbers as the number the running totals get closer and closer to — the limit of the partial sums. Here that limit is $1$, so

$$\frac12 + \frac14 + \frac18 + \dots = 1 .$$

The Greek philosopher Zeno of Elea used puzzles like this one, about 2,500 years ago, to argue that motion is impossible. The modern answer — limits, and the real numbers that guarantee limits exist — took until the nineteenth century to make precise.

Not every infinite sum is finite. The harmonic series $1 + \tfrac12 + \tfrac13 + \tfrac14 + \dots$ grows without bound, even though its terms shrink to zero; Nicole Oresme proved this in the fourteenth century by grouping its terms. Telling these cases apart is the beginning of analysis.