Halfway to the wall · The Lumenary

1

The question

Walk halfway to a wall. Then walk half of what is left, then half of that, and so on, forever. Do you ever reach the wall? How can infinitely many steps add up to a finite distance?

Try the experiment before reading on. There’s no wrong way to explore.

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Experiment

Series and partial sums

Add the terms one by one. Do the partial sums Sₙ settle down, or keep growing?

Examples

Term aₙ of the series

a_n =

partial sums Sₙ
terms aₙ

Partial sums up to NN = 30

S₃₀ = a₁ + … + a₃₀ = 1

Partial sums on a screen are evidence, never proof: a series can look settled and still diverge, as the harmonic series does.

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Hints, if you want them

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What’s going on

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.

Computationally testedAI-drafted · awaiting human review

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The question

Experiment

Hints, if you want them

What’s going on

Where this leads