Practical methods for studying monotonicity

The quotient method for positive sequences

Why use a quotient?

When all the terms of a sequence (u_n) are strictly positive (or all strictly negative), we can examine the sign of u(n+1) – u_n by comparing the quotient u(n+1)/u_n to the number 1 instead. This is often simpler when u_n contains powers, factorials or products.

The rule

For a sequence with strictly positive terms:

if u(n+1)/u_n > 1 for all n, the sequence is strictly increasing;
if u(n+1)/u_n < 1 for all n, the sequence is strictly decreasing;
if u(n+1)/u_n = 1 for all n, the sequence is constant.

Example: geometric sequence

Let u_n = u_0 * q^n, where u_0 > 0 and q > 0. Then u(n+1)/u_n = q. We simply compare q with 1: q > 1 gives an increasing sequence, whilst 0 < q < 1 gives a decreasing sequence.

Example involving powers

Let $u_n = (n+1)/2^n$ for $n$ a natural number ( $u_n > 0$ for all $n$ ). We calculate: u(n+1)/u_n = [(n+2)/2^(n+1)] / [(n+1)/2^n] = (n+2)/(2*(n+1)).

However, (n+2) < 2*(n+1) = 2n+2 implies 0 < n; therefore, as soon as n ≥ 1, the quotient is strictly less than 1: the sequence is strictly decreasing from rank 1 (with u_0 = u_1 = 1).

Common pitfall