If the limit of a sequence equals infinity as $n$ approaches infinity does the sequence diverge?

Question

What I know is that the limit of a sequence is a specific number the sequence converges. I am okay with this. And also I know if the limit does not exist then the sequence diverges. An example of this is an alternating sequence that flips from $1$ to $-1$.

But now the article does not mention if the limit of the sequence is infinity. Should I assume that in this case technically the limit does not exist and the sequence diverges?

Answer 1

We usually use the following terminology:

A sequence:

• converges iff it has a finite limit.
• diverges to infinity iff it has no real limit, however $$\lim_{n\to+\infty}a_n=\pm\infty$$
• diverges if it is not one of the above cases.

This, I think is the clearest way. However, it depends on the context you are using sequences - e.g. you may not mind if a limit is infinite in the case of approximating a Lebesgue-measurable function.

Answer 2

Usually we'd say that the sequence diverges. That's because usually we're considering sequences in $\mathbb{R}$ (the real numbers), and $\infty$ is not a real number. So the sequence has no limit in the real numbers; that means it diverges.

However, it should be clear intuitively that this sequence diverges "in a different way" than an oscillating sequence like $(-1, 1, -1, 1, \dots)$.

Answer 3

As with all things in mathematics, this depends deeply on how the definition is written. That is, the answer to the question of whether or not a sequence "diverges" depends on how you define the term. The typical definition (for sequences of real numbers) is something like

Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of real numbers. We say that $(a_n)$ converges to a limit $L\in\mathbb{R}$, denoted $$ \lim_{n\to\infty} a_n = L $$ if for all $\epsilon > 0$ there exists some $N$ so large that $|a_n - L| < \varepsilon$ whenever $n > N$. If $(a_n)$ does not converge, then we say that it diverges.

In this definition, we are explicitly stating that the limit must be a real number, therefore finite. Therefore if $\lim_{n\to\infty} a_n = \pm\infty$, then the sequence $(a_n)$ diverges. However, the way in which such a series diverges is different from the way that (for example) $((-1)^n)_{n\in\mathbb{N}}$ diverges. Thus it is helpful to have different vocabulary for dealing with such limits. For example:

A sequence $(a_n)_{n\in\mathbb{N}}$ can

• converge as per the above definition,
• diverge to $+\infty$ if for all $M>0$ there exists some $N>0$ such that $a_n > M$ for all $n> N$ (we denote this by writing $\lim\limits_{n\to\infty} a_n = +\infty$),
• diverge to $-\infty$ if for all $M>0$ there exists some $N>0$ such that $a_n < -M$ for all $n> N$ (we denote this by writing $\lim\limits_{n\to\infty} a_n = -\infty$), and
• simply diverge (or, perhaps, oscillate) if it neither converges nor diverges to infinity.

It should also be noted that this definition of convergence is fairly specific to real valued sequences (or, perhaps more generally, sequences coming from noncompact metric spaces). In other mathematical contexts, it is reasonable to say that a sequence converges to $\infty$. For example:

• In measure theory, we are often interested in the extended real line, which is obtained by "attaching" points $-\infty$ and $+\infty$ to the "ends" of the real line. This is typically done so that every set of real numbers has a well-defined supremum and infimum that live in the space under consideration. It solves some minor technical problems, and makes it reasonable to say that a sequence "converges to infinity."
• In complex analysis, the study of linear fractional transformations (among other things) is greatly simplified by attaching a single point to the complex plane, and calling that point $\infty$. Again, this makes it reasonable to talk about sequences that "converges to infinity."
• More generally, any topological space can be "compactified" by attaching a point and making some appropriate identifications. Indeed, the previous example is the one-point compactification of $\mathbb{C}$. In such a compactified space, it makes sense for sequences to converges to infinity.

The basic moral of the story is that sequences (in arbitrary topological spaces) converge to a limit if and only if it "makes sense" to write $$ \lim_{n\to \infty} a_n = L, $$ where $L$ is an element of the space where the sequence lives. Typically, $\pm\infty$ are not elements of our spaces, but can be usefully used to express a particular kind of divergence. However, it is possible to modify our spaces so that infinities are included, thereby giving meaning to the idea of "convergence to infinity."