22nd Putnam 1961

Find lim_n→∞ ∑₁^N n/(N + i²), where N = n².

Solution

As usual, we try integration. We can write the sum as 1/n ∑₁^N 1/(1 + (i/n)²). This is a Riemann sum for the integral ∫₀ⁿ 1/(1 + x²) dx and hence tends to ∫₀^∞ 1/(1 + x²) dx = tan^-1x |₀^∞ = π/2.

22nd Putnam 1961

© John Scholes
jscholes@kalva.demon.co.uk
15 Feb 2002