Oftentimes, one needs to estimate differences of powers of real numbers. The following inequalities are useful for this purpose: $$ n (u-v) v^{n-1} < u^n - v^n < n (u-v) u^{n-1} $$ $$ n x \le (1 + x)^n - 1 $$ $$ (1 + x)^n - 1 \le { n x \over 1 - (n - 1) x} $$

Here $n$ is an integer greater than 1. The first inequality holds when $0 < v < u$ , the second inequality holds when $-1 < x $ , and the third inequality holds when $-1 < x < 1/(n-1)$ . Equality can only occur in the latter two inequalities when $x = 0$ .