Find polynomials p(x) and q(x) with integer coefficients such that p(x)/q(x) = ∑_{0}^{∞} x^{2n}/(1 - x^{2n+1}) for x ∈ (0, 1).

**Solution**

It is an easy induction that the sum of the first n terms is:

(x + x^{2} + ... + x^{2n-1})/(1 - x^{2n}).

But that may be written as [ (1 - x^{2n})/(1 - x) - 1 ] /(1 - x^{2n}) = 1/(1 - x) - 1/(1 - x^{2n}). Since x ∈ (0, 1), the second term tends to 1 as n → ∞. So the result is 1/(1 - x) - 1 = x/(1 - x).

© John Scholes

jscholes@kalva.demon.co.uk

30 Nov 1999