1.2 Tests for Convergence

Now that we have the idea of absolute convergence, we can state some convergence tests applicable to series.

1.2.1 Comparison Test

The first of these tests involves comparing the sizes of two series, and is aptly known as the comparison test.

Theorem 1.4: (Comparison Test)

Let \((a_n)_n\) and \((b_n)_n\) be real sequences, and suppose that there exists a \(M \in \mathbb{N}\) such that \(\lvert a_n \rvert \leq b_n \;\forall n \geq M.\) Then, if \(\sum_{n = 1}^{\infty} b_n\) is convergent, \(\sum_{n = 1}^{\infty} a_n\) is convergent.

Naturally, using this, we can also build a test for divergence to \(\infty\) out of the comparison test too.

Corollary 1.1:

Let \((a_n)_n\) and \((b_n)_n\) be real sequences. If there exists a \(M \in \mathbb{N}\) such that \(0 \leq a_n \leq b_n \; \forall n \geq M\), and \(\sum_{n = 1}^{\infty} a_n\) diverges, then \(\sum_{n = 1}^{\infty} b_n\) diverges.

Here, we require the \(a_n\) values to be non-negative to force any divergence of \(\sum_{n = 1}^{\infty} a_n\) to be to \(\infty\). If we allowed, say, \(a_n = (-1)^nn\), then \(\sum_{n = 1}^{\infty} a_n\) would diverge without limit, making this divergence test useless.

1.2.2 D’Alembert’s Ratio Test

This one is quite similar to the growth factor test for sequences, except that due to the idea of absolute convergence (and Proposition 1.1), the terms of the series only have to be non-zero:

Theorem 1.5: (D’Alembert’s Ratio Test)

Let \((a_n)_n\) be a real sequence with \(a_n \neq 0 \; \forall n \in \mathbb{N}\). Suppose \[\lim_{n\to\infty}\frac{\lvert a_{n+1}\rvert}{\lvert a_n\rvert} = r.\] Then:

  • If \(0 \leq r < 1\), \(\sum_{n = 1}^{\infty} a_n\) converges.
  • If \(r > 1\), then \(\sum_{n = 1}^{\infty} a_n\) diverges.
  • If \(r = 1\), the test is inconclusive.

To see why the test fails for \(r = 1\), consider the three series: \[\sum_{n = 1}^{\infty} \frac{(-1)^{n+1}}{n^2}, \quad \sum_{n = 1}^{\infty} \frac{(-1)^{n+1}}{n} \;\; \text{and} \;\; \sum_{n = 1}^{\infty} (-1)^{n+1}.\] The first is absolutely convergent, the second is conditionally convergent and the third diverges without any limit at all!