Find polynomials p(x) and q(x) with integer coefficients such that p(x)/q(x) = ∑0∞ x2n/(1 - x2n+1) for x ∈ (0, 1).
Solution
It is an easy induction that the sum of the first n terms is:
(x + x2 + ... + x2n-1)/(1 - x2n).
But that may be written as [ (1 - x2n)/(1 - x) - 1 ] /(1 - x2n) = 1/(1 - x) - 1/(1 - x2n). 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