# Divergence to infinity – Serlo

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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 ${\displaystyle +\infty }$ or ${\displaystyle -\infty }$ (so we can assign a formal limit to them) and some don't.

## Motivation

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

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

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

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

## Definition

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

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

Definition (Divergence to ${\displaystyle +\infty }$)

A sequence ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$ diverges to ${\displaystyle +\infty }$, if for any ${\displaystyle S\in \mathbb {R} }$ almost all sequence elements are greater or equal to ${\displaystyle S}$ . That means, for all ${\displaystyle S}$ , there is an index ${\displaystyle N}$, such that ${\displaystyle a_{n}\geq S}$ for all ${\displaystyle n\geq N}$. In quantifier notation:

${\displaystyle \forall S\in \mathbb {R} \,\exists N\in \mathbb {N} \,\forall n\geq N:a_{n}\geq S}$

We can translate this quantifier notation piece by piece:

${\displaystyle {\begin{array}{l}\underbrace {{\underset {}{}}\forall S\in \mathbb {R} } _{{\text{For any real }}S}\ \underbrace {{\underset {}{}}\exists N\in \mathbb {N} } _{{\text{ there is a minimal index }}N,}\ \underbrace {{\underset {}{}}\forall n\geq N:} _{{\text{such that for all indices }}n\geq N}\\[0.5em]\quad \quad \underbrace {{\underset {}{}}a_{n}\geq S} _{{\text{ the element }}a_{n}{\text{ is bigger or equal }}S}\end{array}}}$

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

Definition (Divergence to ${\displaystyle -\infty }$)

A sequence ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$ diverges to ${\displaystyle -\infty }$, if for any ${\displaystyle s\in \mathbb {R} }$ almost all sequence elements are lower or equal to ${\displaystyle s}$ . That means, for all ${\displaystyle s}$ , there is an index ${\displaystyle N}$, such that ${\displaystyle a_{n}\leq S}$ for all ${\displaystyle n\geq N}$. In quantifier notation:

${\displaystyle \forall s\in \mathbb {R} \,\exists N\in \mathbb {N} \,\forall n\geq N:a_{n}\leq s}$

## Notation

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

${\displaystyle \lim _{n\to \infty }a_{n}=+\infty }$

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

${\displaystyle \lim _{n\to \infty }a_{n}=-\infty }$

## Examples

Example

{\displaystyle {\begin{aligned}\lim _{n\to \infty }n&=+\infty \\[0.5em]\lim _{n\to \infty }n^{2}&=+\infty \\[0.5em]\lim _{n\to \infty }-n&=-\infty \end{aligned}}}

Example (geometric sequence)

We have seen in the article "unbounded sequences diverge" that the geometric sequence ${\displaystyle \left(q^{n}\right)_{n\in \mathbb {N} }}$ diverges for ${\displaystyle |q|>1}$. The reason is that for any ${\displaystyle q>1}$ and any real number ${\displaystyle S}$ , the inequality ${\displaystyle q^{n}>S}$ holds for almost all ${\displaystyle n\in \mathbb {N} }$ . Therefore, ${\displaystyle \lim _{n\to \infty }q^{n}=\infty }$ for ${\displaystyle q>1}$. However, if ${\displaystyle q\leq -1}$, there is neither a divergence to ${\displaystyle +\infty }$ nor to ${\displaystyle -\infty }$: For even ${\displaystyle n}$ , the sequence element ${\displaystyle q^{n}}$ is positive and for ${\displaystyle n}$ , it is negative. Hence, the geometric sequence ${\displaystyle \left(q^{n}\right)_{n\in \mathbb {N} }}$ cannot diverge for ${\displaystyle q\leq -1}$ . For ${\displaystyle |q|<1}$ and ${\displaystyle q=1}$, it is convergent.

## Improper convergence

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

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

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

Warning

Some sequences are called "improperly convergent". Despite their name, they are actually divergent sequences. Some limit theorems still hold for improperly divergent sequences, for instance the product rule ${\displaystyle \lim _{n\to \infty }(a_{n}\cdot b_{n})=\lim _{n\to \infty }a_{n}\cdot \lim _{n\to \infty }b_{n}}$ in case both sequences diverge. If one sequence converges and a second one diverges, some rules may also turn out wrong. For instance,

${\displaystyle 1=\lim _{n\to \infty }\left(n\cdot {\frac {1}{n}}\right)=\lim _{n\to \infty }n\cdot \lim _{n\to \infty }{\frac {1}{n}}=\infty \cdot 0=0}$

Applying the product rule to a product of a convergent and divergent sequence leads to the wrong result ${\displaystyle 1=0}$. Always be careful, when using limit theorems for improperly convergent sequences! In the article "Divergence to infinity: rules" we will find out, which rules for limit calculations still hold for improperly convergent sequences.