Putnam 1992

Let the coefficient of x¹⁹⁹² in the power series (1 + x)^α = 1 + αx + ... be C(α). Find ∫₀¹ C(-y-1) ∑_k=1¹⁹⁹²1/(y+k) dy.

Solution

Trivial.

The usual Taylor series gives C(-y-1) = (y+1)(y+2) ... (y+1992)/1992! . Hence the integrand is simply the derivative of (y+1)(y+2) ... (y+1992)/1992! , so the integral evaluates to 1993 - 1 = 1992.

Putnam 1992

© John Scholes
jscholes@kalva.demon.co.uk
24 Dec 1998