Show that for any integer n > 1, we have 1/e - 1/(ne) < (1 - 1/n)^{n} < 1/e - 1/(2ne).

**Solution**

Take the left-hand inequality first. Multiplying through by e and taking logs, we want to show that 0 < 1 + (n-1) ln(1 - 1/n). Putting x = 1/n, it is sufficient to show that ln(1 - x) > x/(1-x) for 0 < x < 1. But we can use the expansions: ln(1 - x) = -(x + x^{2}/2 + x^{3}/3 + ... ), and x/(1-x) = x + x^{2} + x^{3} + ... and the result is immediate.

Similarly, for the right-hand inequality we need: 1 + n ln(1 - 1/n) < ln(1 - 1/2n). Putting x = 1/n, it is sufficient to show that -(x/2 + x^{2}/3 + x^{3}/4 + ... ) < -(x/2 + (x/2)^{2}/2 + (x/2)^{3}/3 + ... ) for 0 < x < 1. Again this holds term by term because n2^{n} > n+1 for n > 1.

© John Scholes

jscholes@kalva.demon.co.uk

11 Dec 2002