a_{n} is a sequence of positive reals. Show that lim sup_{n→∞}( (a_{1} + a_{n+1})/a_{n})^{n} ≥ e.

**Solution**

It is sufficient to show that (a_{1} + a_{n+1})/a_{n} ≥ 1 + 1/n for infinitely many n. For these n then form a sequence for which ( (a_{1} + a_{n+1})/a_{n} )^{n} has a limit point (possibly infinity) not less than lim (1 + 1/n)^{n} = e.

If that were false, then we would have (a_{1} + a_{n+1})/a_{n} < 1 + 1/n for all n not less than some N. In particular, taking n = N, N+1, N+2, ... we get a_{1}/N+1 + a_{N+1}/N+1 < a_{N}/N, a_{1}/N+2 + a_{N+2}/N+2 < a_{N+1}/N+1, a_{1}/N+3 + a_{N+3}/N+3 < a_{N+2}/N+2, ... . Hence a_{n}/N > a_{1}(1/N+1 + 1/N+2 + ... + 1/N+M) + a_{N+M}/N+M > a_{1}(1/N+1 + 1/N+2 + ... + 1/N+M). But this holds for any M and (1/N+1 + 1/N+2 + ... ) diverges. Contradiction.

© John Scholes

jscholes@kalva.demon.co.uk

11 Mar 2002