The sum that never stops growing

Add 1, then a half, then a third, a quarter, a fifth, and keep going. Every term is smaller than the last, and the terms head towards zero. Yet the total never settles on any number. Pick any target, a hundred or a million, and the running total passes it eventually.

Shrinking is not enough

This sum is called the harmonic series. Write H_n for its running total after n terms, the partial sum. Kifowit and Stamps, in a 2006 survey of twenty proofs, open with why it matters: few series show more clearly that terms going to zero does not make a sum converge, meaning settle on a finite total.

That runs against intuition. If each new piece is tiny, surely the total creeps towards some ceiling? For many series it does. For this one the pieces shrink too slowly. No term is ever zero, and there are always enough of them left to add up to something that matters.

Oresme's grouping

The oldest proof in the survey is Nicole Oresme's, from about 1350. It needs only one trick: group the terms in blocks that double in length.

After the 1, the first block is just 1/2. The next is 1/3 + 1/4. Each of those is at least 1/4, so the pair adds to more than 1/2. The block after that is 1/5 through 1/8: four terms, each at least 1/8, so again more than 1/2. The next block has eight terms, each at least 1/16. The pattern never breaks: the block that ends at 1/2^j has 2^j−1 terms, none smaller than 1/2^j, so it adds to more than a half whenever it has two or more terms.

So after 2^k terms the total is at least 1 + k/2. Every doubling of the number of terms adds at least another half. Double often enough and the total passes any number you name. That is what divergence means: the partial sums are unbounded.

How fast it grows

Unbounded does not mean fast. Our own arithmetic: the first ten terms add to about 2.93, a hundred terms to 5.19, a thousand to 7.49, and a million terms to only 14.39.

Those numbers follow a pattern Euler pinned down in 1731. Lagarias's history of the result gives it: H_n minus the natural logarithm of n approaches a fixed number, now called Euler's constant, γ = 0.57721…. In our computation the gap is 0.626 at ten terms, 0.582 at a hundred, 0.578 at a thousand and 0.57722 at a million. So the running total tracks ln n + γ, with corrections that fade as n grows.

Turn that round and you get a figure we derive here from Euler's own form, which neither source states outright: to push the total past a target T takes about e^T−γ terms. Counting directly, passing 10 takes 12,367 terms; passing 15 takes 1,835,421. Each extra unit of total costs about 2.7 times as many terms as the last.

In short

The terms of 1 + 1/2 + 1/3 + … tend to zero, but the sum does not converge. Oresme's grouping shows why: split the terms into blocks that double in length and every block beyond the first is worth more than a half, so the total climbs past any bound. It climbs like the logarithm of the number of terms, which is why the growth is real and yet almost invisible.