1.1 Series Convergence

Recall from last week that we can define the convergence of an infinite sum/series as follows:

Definition 1.1: (Series Convergence and Partial Sums)
Let \((a_n)_{n \in \mathbb{N}}\) be a real sequence. Then \(\sum_{n = 1}^{\infty} a_n\) converges if and only if the sequence \((S_N)_{N \in \mathbb{N}}\) converges, where \[S_N:= \sum_{n = 1}^{N} a_n\] is the \(N^{\text{th}}\) partial sum. If \(S_N \to \ell\) as \(N \to \infty\), we define \[\ell = \sum_{n = 1}^{\infty}a_n.\]

Much like with proving sequence convergence, using the definition each time you want to `evaluate’ a series can get tedious really quickly. Therefore, we really want a couple of tests which can prove convergence without too much hassle. Before we discuss these tests though, we need to introduce the ideas of absolute and conditional convergence.

Definition 1.2: (Absolute Convergence)
A real series \(\sum_{n = 1}^{\infty} a_n\) is absolutely convergent if \(\sum_{n = 1}^{\infty} \lvert a_n \rvert\) converges.

For example, if we consider the series \(\sum_{n = 1}^{\infty} a_n\), where \(a_n\) is given by \[a_n = \frac{(-1)^n}{n^2},\] we find that \[\sum_{n = 1}^{\infty} \lvert a_n \rvert = \sum_{n=1}^{\infty} \frac{1}{n^2},\] which we know converges from lectures1. Hence \(\sum_{n = 1}^{\infty} a_n\) is absolutely convergent. Have we learnt anything about the convergence of \(\sum_{n=1}^{\infty}a_n\) here? Turns out the answer is yes, and this is because of the following result.

Proposition 1.1:

If a real series \(\sum_{n = 1}^{\infty} a_n\) is absolutely convergent, then it is convergent.

At this stage, we can introduce the idea of conditional convergence too.

Definition 1.3: (Conditional Convergence)
Let \(\sum_{n = 1}^{\infty} a_n\) be a real series. If \(\sum_{n = 1}^{\infty} a_n\) is convergent, but \(\sum_{n = 1}^{\infty} \lvert a_n \rvert\) is not, then \(\sum_{n = 1}^{\infty} a_n\) is said to be conditionally convergent.

1.1.1 Series Rearrangement

So, what can we do with absolutely convergent series?
Theorem 1.2:

Suppose \(\sum_{n = 1}^{\infty} a_n\) is an absolutely convergent series, and that \(\sigma: \mathbb{N} \to \mathbb{N}\) is a bijection. Then \(\sum_{n = 1}^{\infty} a_{\sigma(n)}\) is also an absolutely convergent series, and \[\sum_{n = 1}^{\infty} a_n = \sum_{n = 1}^{\infty} a_{\sigma(n)}.\]

This theorem tells us that for an absolutely convergent series, we can order the terms any way we like, and still reach the same value for the series. At this point, you might be interested to know what happens if we don’t have absolute convergence. Long story short, weird things can happen, as is seen in the following theorem.

Theorem 1.3: (Riemann Rearrangement Theorem)

Suppose \(\sum_{n = 1}^{\infty} a_n\) is conditionally convergent. Then, for any \(\alpha \in \mathbb{R}\), or \(\alpha = \pm\infty\), there exists a bijection \(\sigma:\mathbb{N} \to \mathbb{N}\) such that \[\sum_{n = 1}^{\infty} a_{\sigma(n)} = \alpha.\]

So what we see here is that we really need to be careful in which order we sum up the terms of a conditionally convergent series!

  1. If you take the Vector Calculus and PDEs module next year, you’ll show that this sum equals \(\frac{\pi^2}{6}\).↩︎