Stacking with harmonic series
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How long can dominoes (or bricks, tiles etc.) stack be? Can it be 50 cm? 1m? 1 km? Let’s take a look:
For each of the stacks above we need to make sure that it does not collapse. Continuing in the same way for [math]\displaystyle{ n }[/math] dominos, the first overhanging by 1/2, the second by 1/4, the third by 1/6,..., the [math]\displaystyle{ n }[/math]-th by [math]\displaystyle{ 1/2n }[/math], we get the total length of [math]\displaystyle{ 1/2 +1/4+1/6+\dots +1/2n = \frac{1}{2}(1+1/2+1/3+…+1/n) }[/math]. Since the harmonic series diverges, we can get any length as soon as we have enough dominos!