Let α be a positive real. Let an = S1n (α/n + i/n)n. Show that lim an ∈ (eα, eα+1).
The largest term is (1 + α/n)n which tends to eα. The next largest is (1 + (α - 1)/n)n which tends to eα - 1 and so on. All terms are positive, so the limit is at least eα + eα - 1 > eα. Also the limit is at most eα(1 + 1/e + 1/e2 + ... ) < eα(1 + 1/2 + 1/4 + ... ) = 2eα < eα + 1.
14th Putnam 1954
© John Scholes
4 Dec 1999