Two Infinite Sums, Opposite Fates

Halving the gap

Add $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\dots$1/2 + 1/4 + 1/8 + 1/16 + …, where each term is half the one before, and keep going forever. The running totals, called partial sums, are $\frac{1}{2}$1/2, $\frac{3}{4}$3/4, $\frac{7}{8}$7/8 and $\frac{15}{16}$15/16. Each one is short of 1 by exactly the last term added: $\frac{1}{2}$1/2, then $\frac{1}{4}$1/4, then $\frac{1}{8}$1/8, then $\frac{1}{16}$1/16.

So after n terms the total is $1-\frac{1}{2^n}$1 − 1/2ⁿ. Every new term fills half of the gap that is left, so the gap halves each time and never closes completely. The partial sums never pass 1, and they get as close to 1 as you like: after 10 terms the gap is $\frac{1}{1024}$1/1024, and after 20 terms it is less than one millionth. The sum to infinity is exactly 1.

A sum that closes in on one number in this way converges, and that number is its sum.

The partial sums $\frac{1}{2}$1/2, $\frac{3}{4}$3/4 and $\frac{7}{8}$7/8 on a number line. Each step up is half the one before, and the totals close in on 1.

A square of area 1. The first term colors half of it, and each term after that colors half of what is left. After 3 terms $\frac{7}{8}$7/8 is colored and $\frac{1}{8}$1/8 is left. Drag the number of terms up to 10: the gap shrinks to $\frac{1}{1024}$1/1024 in one corner.

The harmonic series

Now add the reciprocals of the whole numbers: $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\dots$1 + 1/2 + 1/3 + 1/4 + 1/5 + …. This is the harmonic series. Every term is smaller than the one before, and the terms shrink toward 0, just as the halving terms did.

The first eight partial sums, to 3 decimal places, are 1, 1.5, 1.833, 2.083, 2.283, 2.45, 2.593 and 2.718. They grow, and each step is smaller than the one before. From those numbers alone, it looks as if they might settle on something near 3.

The first four partial sums of the harmonic series, 1, 1.5, 1.83 and 2.08 to 2 decimal places. They have already passed 2.

Group the terms

Group the terms into blocks, each twice as long as the one before. After 1 and $\frac{1}{2}$1/2, the next block is $\frac{1}{3}+\frac{1}{4}$1/3 + 1/4. Both terms are at least $\frac{1}{4}$1/4, so $\frac{1}{3}+\frac{1}{4}>\frac{1}{4}+\frac{1}{4}=\frac{1}{2}$1/3 + 1/4 > 1/4 + 1/4 = 1/2.

The next block is $\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}$1/5 + 1/6 + 1/7 + 1/8: four terms, each at least $\frac{1}{8}$1/8, so their sum is more than $4\times \frac{1}{8}=\frac{1}{2}$4 × 1/8 = 1/2. The block after that runs from $\frac{1}{9}$1/9 to $\frac{1}{16}$1/16: eight terms, each at least $\frac{1}{16}$1/16, so it adds more than $8\times \frac{1}{16}=\frac{1}{2}$8 × 1/16 = 1/2. Every later block has twice as many terms, each at least half as big, so each block adds more than $\frac{1}{2}$1/2 to the total.

So the first $2^k$2ᵏ terms add up to at least $1+\frac{k}{2}$1 + k/2. Check it against the true sums: 4 terms give 2.083, at least 2; 8 terms give 2.718, at least 2.5; 16 terms give 3.381, at least 3. Taking enough blocks passes 10, 100, or any number you name, so the harmonic series grows without limit.

The partial sums after 2, 4, 8 and 16 terms: 1.5, 2.08, 2.72 and 3.38, to 2 decimal places. The blocks $\frac{1}{3}$1/3 to $\frac{1}{4}$1/4, $\frac{1}{5}$1/5 to $\frac{1}{8}$1/8 and $\frac{1}{9}$1/9 to $\frac{1}{16}$1/16 add about 0.58, 0.63 and 0.66, each more than $\frac{1}{2}$1/2.

Converge or diverge

A sum that grows without limit diverges. The harmonic series diverges, but very slowly: after 1000 terms the total is only about 7.49, and after a million terms about 14.39. A calculator adding terms one by one would suggest it settles. The grouping shows that it does not.

So terms that shrink to 0 are needed for a sum to converge, but they are not enough. The halving terms shrink fast, each one half of the last, and their sum converges to 1. The harmonic terms shrink so slowly that every block of them still adds more than $\frac{1}{2}$1/2, and their sum diverges.

The usual mistakes

Thinking an endless sum must be infinite. The partial sums of $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots$1/2 + 1/4 + 1/8 + … are all below 1, and the sum is exactly 1.

Giving 2 for the halving series. The sum $1+\frac{1}{2}+\frac{1}{4}+\dots$1 + 1/2 + 1/4 + … is 2, because it starts with an extra 1. Starting from $\frac{1}{2}$1/2, the sum is 1.

Thinking shrinking terms always give a finite sum. The terms of $1+\frac{1}{2}+\frac{1}{3}+\dots$1 + 1/2 + 1/3 + … shrink to 0, but blocks of them keep adding more than $\frac{1}{2}$1/2, so the sum passes every number.

Trusting a few partial sums. The harmonic series looks as if it is settling near 3 after eight terms, but it passes 4 after 31 terms and keeps growing.

A bouncing ball and a stack of books

In the application below, a ball bounces to half its height each time, and its total distance is a halving sum that converges. A stack of books leans out over a table edge by half the harmonic series, so the stack can reach as far out as you like.