Stacked Books Can Reach Any Distance

Stack books at the edge of a table, each pushed a little further out than the one below. With just four books the top one hangs completely past the edge, and with enough books you can reach as far as you like: no glue, just a sum that never stops growing. A stack stands while the balance point of everything above each book sits over that book. The top book can stick out 1/2 of its length, the next 1/4, then 1/6, 1/8: four books reach 25/24 of a book length. The reach is half of the harmonic series 1 + 1/2 + 1/3 + ..., which grows painfully slowly (about 5.19 after 100 terms, 14.39 after a million, about 230.8 after a googol; it stays close to ln n + 0.5772, the Euler-Mascheroni constant) but never stops: Nicole Oresme's proof (c. 1350) groups the terms into halves. Two book lengths need 31 books, three need 227, ten about 272 million. The same growth lets an ant on a stretching rubber rope reach the end. Compare 1 + 1/4 + 1/9 + ..., which settles at pi^2/6. Kempner (1914): drop every term with a 9 and the rest sums to about 22.92. Paterson and Zwick (2006) showed that books side by side, with counterweights, reach like the cube root of the number of books, which is the best possible up to a constant. 0:00 Books past the table 0:24 The balance rule 0:41 Half, quarter, sixth, eighth 1:05 Four books: 25/24 1:20 The harmonic series 1:34 Painfully slow 1:49 The logarithm 2:05 Oresme's proof 2:31 The price 2:47 The ant on the rubber rope 3:10 Basel, by contrast 3:27 No nines allowed 3:51 Better stacks #harmonicseries #physics #math