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By Root 583 0

numbers. The harmonic series looks like it should converge. Each term in the series gets smaller and smaller, so you would think that the sum of all the terms would be bounded by a fixed amount. But, bizarrely, the harmonic series is divergent, a decelerating but unstoppable snail. After 100 terms of the series, the total has only just passed 5. After 15,092,688,622,113,788,323,693,563,264,538, 101,449,859,497 terms, the total exceeds 100 for the first time. Yet this stubborn snail will continue its bid for freedom, past any distance you care to mark out. The series will eventually reach a million, then a billion, going further and further towards infinity. (The proof is included as an appendix.)

The harmonic series appears when we consider the maths of stacking Jenga blocks. Say you have two blocks and want to position them one on top of the other so that the top one has the largest possible overhang, but doesn’t topple over. The way to do this is to place the top block exactly halfway across the one underneath, as demonstrated opposite (A). In this way, the centre of gravity of the top block falls on the edge of the bottom brick.

If we had three blocks, what would be their positions so the combined overhang was as large as possible without toppling? The solution is for the top one to be placed halfway along the middle one, and for the middle one to be a quarter of the way along the bottom one, as in the diagram opposite (B).

Continuing for more and more blocks, the general pattern is that, in order to guarantee the maximum combined overhang, the top one must be halfway along the second one, which must be a quarter along the third one, which must be a sixth along the fourth one, which must be an eighth along the fifth one, and so on. This gives us a leaning tower of bricks that looks like C on the page opposite.

How to stack Jenga blocks with maximum overhang so that they don’t topple.

The total overhang of this tower, which is the sum of all the individual overhangs, is the following series:

Which can be rewritten as:

Which is half of the harmonic series, if we carry on for an infinite number of terms.

Now, since we know that the harmonic series increases to infinity, we also know that the harmonic series divided by two increases to infinity, because infinity divided by two is infinity. Rephrased in the context of stacking Jenga blocks, this means that it is theoretically possible to create a freestanding overhang of any length we want. If the harmonic series divided by two will eventually exceed any number we want, provided we include enough terms, then the overhang of the leaning tower of blocks will eventually exceed any length we want, provided we stack enough blocks. Although theoretically possible, however, the practicalities of constructing a tower with a large overhang are daunting. In order to achieve an overhang of 50 blocks, we would need a tower of 15×1042 blocks – which would be much higher than the distance from here to the edge of the observable universe.

The delights of the harmonic series are profuse, so let’s have some more fun with it. Consider the harmonic series excluding every term that has a 9 in it, which is also an infinite series. In other words, we are extracting the following terms:

So, the depleted series looks like:

Recall that the harmonic series adds up to infinity, so one might think that the harmonic series with no 9s also adds up to a pretty high number. Wrong. It adds up to just under 23.

By filtering out 9 we have tamed infinity: we have slaughtered the beast of eternity and all that is left is a shrivelled carcass of about 23.

This result appears remarkable, but by looking a little closer we can understand it completely. Eliminating a 9 gets rid of only one of the first 10 terms of the harmonic series. But it gets rid of 19 of the first 100 terms, and 271 of the first thousand. By the time the numbers are very large, sayrhang are100 digits, the vast majority of numbers contain a 9. As it turns out, thinning the harmonic series by taking out the