# Using your Head is Permitted

## April 2008 solution

The answer is "infinity".
With an infinite number of bins, you can compose the entire harmonic series,
which is known to diverge.

For any finite number of bins, there is a finite number of primes that compose
all the elements in the sub-series. Consider the product of
*p*/(*p*-1) over all *p* in this finite set of primes. The
value of this product is clearly finite. Note, however, that

*p*/(*p*-1)=1+1/*p*+1/*p*^{2}+1/*p*^{3}+...

Though proving the point rigorously is not very elegant, it is nevertheless
correct (and intuitively clear) that multiplying *p*/(*p*-1) for
different *p* values is the same as multiplying the geometric series that
sum to the same value and (by distributivity) to the sum of the products
of elements chosen one from each geometric series, which, in turn, is equivalent
to calculating the desired sum of the partial harmonic series.

Therefore, not only does the series converge for any finite choice of bins,
its limit is explicitly calculable.

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