1.3 Sequences and Convergence

We’re finally onto the first main topic of this course! To discuss anything from this point, we need to introduce the idea of a sequence.
Definition 1.2: (Sequence)
A sequence of real/complex numbers is a function \[\begin{align*} a:\; &\mathbb{N} \longrightarrow X,\\ &n \longmapsto a_n, \end{align*}\] where \(X\) is either \(\mathbb{R}\) or \(\mathbb{C}\) respectively.

Since this notation can get kind of annoying, we instead denote a sequence by \((a_n)_{n\in\mathbb{N}}\). If it’s clear from the context what set we’re indexing over, we can even just simply write a sequence as \((a_n)_n\).

Now, this gives us an infinitely long list of real numbers, and sometimes its interesting to look at the ‘long-term’ behaviour of these lists. This gives rise to the idea of convergence.

Definition 1.3: (Sequence Convergence)
A sequence \((a_n)_{n\in\mathbb{N}}\) converges to a real number \(L\) as \(n \longrightarrow \infty\), written as either \(a_n \longrightarrow L\), or \(\lim_{n \to \infty}a_n = L\) if \[\forall \epsilon > 0, \; \exists N = N(\epsilon) \in \mathbb{N}, \; \text{such that} \; \forall n \geq N, \; \lvert a_n - L \rvert < \epsilon.\]

Loosely speaking, this says that no matter how close you want the sequence to get to \(L\), you will always be able to find some point in the sequence after which all points in the sequence will be as close to \(L\) as you wanted. For an example of this, have a look at this Desmos link. For \(\epsilon = 0.5\) and \(L = 3\), you can see that every member of the sequence after the \(11^{th}\) lies within a strip of width \(2\epsilon\) around \(L\). Have a go at messing with the value of \(\epsilon\)!

Something else we can mention for the definition is its negation. Specifically, a sequence \((a_n)_n\) does not converge to \(L\) if \[\begin{align*} \exists\; \epsilon_0 > 0, \; \text{such that} \; \forall N \in \mathbb{N}, \exists n \geq N \; \text{such that} \; \lvert a_n - L \rvert \geq \epsilon_0. \end{align*}\]

We can say the exact same things for the convergence of complex sequences without much effort too. We just need to remember to change the distance from the absolute value to the modulus!

1.3.1 Useful Sequences

Since there’s no point in having a definition without using it, it’s a great idea to obtain some (straightforward) results:

  • As \(n \longrightarrow \infty\): \[\frac{1}{n} \longrightarrow 0.\]
  • For a real number \(c\): as \(n \longrightarrow \infty\), \[c \longrightarrow c.\]
  • For \(q \in \mathbb{R}\) with \(\lvert q \rvert < 1\): as \(n \longrightarrow \infty\) \[q^n \longrightarrow 0.\]