Math for Non-Geeks/Divergence to infinity

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Until to now, we investigated when and how a sequence diverges. In this chapter, we will investigate divergent series. Not every divergent series behaves the same: some can be seen to diverge to $+\mathrm{\infty }$ or $-\mathrm{\infty }$ (so we can assign a formal limit to them) and some don't.

Motivation

Some sequences do not converge, but they unambiguously tend to $+\mathrm{\infty }$ or $-\mathrm{\infty }$. For instance consider the sequences $a_n=n$, $b_n=2^n$ and $c_n=-n+2\cdot (-1{)}^n$:

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  The sequence $a_n=n$ tends towards $+\mathrm{\infty }$.
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  The sequence $b_n=2^n$ tends towards $+\mathrm{\infty }$ even faster.
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  The sequence $c_n=-n+2\cdot (-1{)}^n$ tends towards $-\mathrm{\infty }$.

We will give a mathematical definition which classifies these sequences as diverging towards $+\mathrm{\infty }$ or $-\mathrm{\infty }$. Some other sequences may not diverge this way. For instance, consider $d_n=(-1{)}^n$ or $e_n=(-1{)}^n\cdot n$ .

The sequence $d_n=(-1{)}^n$ is bounded and can therefore neither tend to $+\mathrm{\infty }$, not to $-\mathrm{\infty }$ .

The sequence $e_n=(-1{)}^n\cdot n$ is unbounded, but contains parts (subsequences) tending towards $+\mathrm{\infty }$ and parts tending towards $-\mathrm{\infty }$ :

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  The alternating sequence $d_n=(-1{)}^n$ is bounded, so it cannot tend towards infinity.
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  The sequence $e_n=(-1{)}^n\cdot n$ is unbounded, but neither tends to $+\mathrm{\infty }$ nor to $-\mathrm{\infty }$.

Definition

We have observed some sequences, which tend towards $+\mathrm{\infty }$ or $-\mathrm{\infty }$ . How can we give a mathematically precise classification for this observation?

Let us start with divergence to $+\mathrm{\infty }$: "A sequence diverges to $+\mathrm{\infty }$" means that it grows larger than any number $S$, no matter how large (since $+\mathrm{\infty }$ is greater than any $S$). Even further, almost all sequence elements must be greater than $S$ . Or equivalently, we need a sequence element number $N$, such that any element coming after it is bigger than $S$ . Indeed, this is already sufficient as condition for divergence to $+\mathrm{\infty }$:

We can translate this quantifier notation piece by piece:

Divergence to $-\mathrm{\infty }$ is just defined analogously:

Notation

For $(a_n{)}_{n\in \mathbb{N}}$ diverging to $+\mathrm{\infty }$ , we write

Analogously, for $(a_n{)}_{n\in \mathbb{N}}$ diverging to $-\mathrm{\infty }$ , we write

Improper convergence

The notation $\underset{n\to \mathrm{\infty }}{\lim }{a}_n=+\mathrm{\infty }$ suggests that $(a_n{)}_{n\in \mathbb{N}}$ kind of "converges" to infinity. This is no convergence in the usual sense, since the symbolic expression $+\mathrm{\infty }$ is not a number. However, the divergence to $\pm \mathrm{\infty }$ has a lot in common with convergence to a number (except for boundedness):

Convergence	Divergence to $\pm \mathrm{\infty }$
In each $ϵ$-neighbourhood (interval), we can find almost all sequence elements.	In each interval $[S,\mathrm{\infty })$, we can find almost all sequence elements.
All subsequences converge to the same limit.	All subsequences also diverge to $\pm \mathrm{\infty }$.
Every convergent sequence is bounded.	Every sequence diverging to $\pm \mathrm{\infty }$ is unbounded.

Especially, some of the limit theorems hold true for $\pm \mathrm{\infty }$ and in some cases, one may threat a sequence diverging to $\pm \mathrm{\infty }$, similar as a convergent sequence. Sometimes, the divergence to $\pm \mathrm{\infty }$ is even called "improper convergence". However, always keep in mind that an an improper convergence is still a divergence.

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