Putnam 99/A4 solution

60th Putnam 1999

Problem A4

Let a_ij = i²j/(3ⁱ(j 3ⁱ + i 3^j)). Find ∑ a_ij where the sum is taken over all pairs of integers (i, j) with i, j > 0.

Solution

Let b_i = i/3ⁱ, then a_ij = b_i²b_j(b_i + b_j). Let the sum be k. Then k = ∑ a_ij = 1/2 ∑ (a_ij + a_ji) = 1/2 ∑ b_ib_j. We have the familiar 1 + 2x + 3x² + ... = 1/(1 - x)², so x + 2x² + 3x³ + ... = x/(1 - x)², and hence ∑ b_i = 1/3 (2/3)^-2 = 3/4. So k = 1/2 (3/4)² = 9/32.

60th Putnam 1999