Difference between revisions of "Stacking with harmonic series"
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| − | How ''long'' can | + | How ''long'' can dominoes (or bricks, tiles etc.) stack be? Can it be 50 cm? 1m? 1 km? Let’s take a look: |
[[File:Domino.png|alt=Domino stacks|none|thumb|777x777px|Domino stacks]] | [[File:Domino.png|alt=Domino stacks|none|thumb|777x777px|Domino stacks]] | ||
| + | For each of the stacks above we need to make sure that it does not collapse. | ||
Continuing in the same way for <math> n </math> dominos, the first overhanging by 1/2, the second by 1/4, the third by 1/6,..., the <math> n </math>-th by <math> 1/2n </math>, we get the total length of <math> 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! | Continuing in the same way for <math> n </math> dominos, the first overhanging by 1/2, the second by 1/4, the third by 1/6,..., the <math> n </math>-th by <math> 1/2n </math>, we get the total length of <math> 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! | ||
Revision as of 20:30, 15 January 2022
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!