Why 1/2 + 1/4 + 1/8 + … Adds Up to Exactly 1

Updated 4 October 2026

Infinitely many halves still add up to exactly 1.

$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots$1/2 + 1/4 + 1/8 + ... to infinity = 1

Why does 1/2 + 1/4 + 1/8 + … add up to exactly 1?

The sum 1/2 + 1/4 + 1/8 + … equals exactly 1. Each step covers half of what is left: after 1/2, the gap to 1 is 1/2, after another 1/4 the gap is 1/4, and after n steps the gap is 1/2ⁿ. The partial sums run 1/2, 3/4, 7/8, 15/16, each one short of 1 by its last term. An infinite sum means the number these partial sums settle on, and the gap shrinks below any positive amount you name: 20 steps bring it under one millionth, because 2 to the 20th power is more than a million. Every number smaller than 1 is passed by some partial sum, and no partial sum passes 1, so the limit is exactly 1. The geometric series formula agrees: a first term a and ratio r sum to a ÷ (1 − r), and ½ ÷ (1 − ½) is ½ ÷ ½, which is 1.

1. Walk half the way to the wall: half the distance is behind you, half ahead.

2. Each step halves what is LEFT. After four steps, only one sixteenth remains.

3. The gap is always $\frac{1}{2^n}$1/2ⁿ, and no number is smaller than every one of those.

All 36 shortcuts: the Magic Tricks library.