In my last question, we came across parts of the so-called harmonic series 1/2 + 1/3 + 1/4 + 1/5 ... 1/n. Now, this series is divergent; it does so very slowly, but as n tends towards infinity so does the sum. However, the sum of the squares of each term is convergent.
So, numerically, if we add together the areas of squares with sides of 1/2 units, 1/3 units and so on, we will tend towards a finite limit. However, is it possible to geometrically fit these squares inside a square of 1 unit side without any of them overlapping?
Just to show that this is not a trivial question, the answer is negative for any Pythagorean triplet. For example, we know that 32 + 42 = 52, but it is not possible to put squares of sides 3 and 4 inside a square of side 5 units without an overlapping region.
So, to restate the question, we have an infinite number of squares whose sides are in the sequence 1/2, 1/3, 1/4, 1/5... 1/n... and we wish to fit them inside a unit square. Is this possible, and if so, what is the packing method?
Now, this led me thinking about whether this is true for other shapes. If we have an equilateral triangle of unit sides, can we pack an infinite number of smaller triangles inside it, whose sides are in the harmonic sequence and without any overlap? Again, if we can, what is the packing method?
And lastly, can we do the same with circles? That is, given a circle of radius 1 unit, we wish to pack inside it an infinite number of circles whose radii are in the harmonic sequence. Again, if this is possible, what is the packing technique?
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