Let α be a positive real. Let a_{n} = S_{1}^{n} (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/e^{2} + ... ) < e^{α}(1 + 1/2 + 1/4 + ... ) = 2e^{α} < e^{α + 1}.

© John Scholes

jscholes@kalva.demon.co.uk

4 Dec 1999