1.2 Series
It might look like we’re done with sequences, but in the grand scheme of things, we’re only really getting started. Since with each sequence \((a_n)_{n\in\mathbb{N}}\), we have an infinite list of real numbers, we might consider trying to manipulate them in some way. One way we can do this is by adding them together, which leads to the notion of a series.
If \((S_N)_{N\in\mathbb{N}}\) diverges to \(\pm\infty\), we say that the corresponding series \[\sum_{n=1}^{\infty} a_n = \pm\infty.\] Finally, if \((S_N)_{N\in\mathbb{N}}\) doesn’t converge to a limit, we say that the series diverges without limit.
1.2.1 Algebra of Series
By applying the algebra of limits to the sequences of partial sums, we can deduce some handy results.
Let \(\sum_{n=1}^{\infty} a_n\) and \(\sum_{n=1}^{\infty} b_n\) be convergent series, and let \(\alpha,\beta \in \mathbb{R}\). Then \[\sum_{n = 1}^{\infty} (\alpha a_n + \beta b_n) = \alpha\sum_{n=1}^{\infty} a_n + \beta\sum_{n=1}^{\infty} b_n.\]
1.2.2 Some Other Useful Results
Firstly, we can relate the size of the terms of a series to the overall sum.
Let \(\sum_{n=1}^{\infty} a_n\) and \(\sum_{n=1}^{\infty} b_n\) be real series. If \(a_n \leq b_n \, \forall n\in\mathbb{N}\), then \[\sum_{n=1}^{\infty} a_n \leq \sum_{n=1}^{\infty} b_n.\]
Secondly, we have a necessary condition for convergence of a series.
Let \(\sum_{n=1}^{\infty} a_n\) be a convergent series. Then \(a_n \to 0\) as \(n \to \infty\).
Note that the converse of this theorem does not hold (think of the sum \(\sum_{n=1}^{\infty} \frac{1}{n}\)). However, the contrapositive is very good at showing that a series does not converge!
Let \(\sum_{n=1}^{\infty} a_n\) be a series. If \(a_n \not\to 0\) as \(n \to \infty\), then \(\sum_{n=1}^{\infty} a_n\) does not converge.