1.1 Tests for Convergence

As you’ve seen, using the definition to prove that sequences converge can get quite tedious! Hence, it’s useful to have some ‘quick’ tests to determine whether sequences converge. This is where the Growth Factor Test (or ratio test) comes in.

Theorem 1.1: (Growth Factor Test)

Let \((a_n)_{n\in\mathbb{N}}\) be a real sequence with \(a_n>0 \; \forall n\in\mathbb{N}\), and with \[\lim_{n\to\infty} \frac{a_{n+1}}{a_n} = r.\] Then:

• If \(r < 1\), \(a_n \to 0\) as \(n \to \infty\).
• If \(r > 1\), \(a_n \to \infty\) as \(n \to \infty\).
• If \(r = 1\), the test is inconclusive.

To see why the test fails for \(r = 1\), consider the sequences \((a_n)\) and \((b_n)\) where \(a_n = n\) and \(b_n = 1/n\). Something to note here is that if the terms of a sequence are defined by a ratio of polynomials, then the growth factor test will be inconclusive (Can you see/prove why?).