14th Putnam 1954

Let α be a positive real. Let a_n = S₁ⁿ (α/n + i/n)ⁿ. Show that lim a_n ∈ (e^α, e^α+1).

Solution

The largest term is (1 + α/n)ⁿ which tends to e^α. The next largest is (1 + (α - 1)/n)ⁿ which tends to e^{α - 1} and so on. All terms are positive, so the limit is at least e^α + e^{α - 1} > e^α. Also the limit is at most e^α(1 + 1/e + 1/e² + ... ) < e^α(1 + 1/2 + 1/4 + ... ) = 2e^α < e^{α + 1}.

14th Putnam 1954

© John Scholes
jscholes@kalva.demon.co.uk
4 Dec 1999