Improperly Convergent Sequence

analysis

Definition

Uneigentlich Konvergente Folge

Improperly Convergent Sequence

A sequence $(a_n{)}_{n\ge 0}$ whose elements become arbitrarily large with:

$$\forall K>0:\exists N(K)\in \mathbb{N}:\forall n>N(K):a_n>K$$

and can be denoted as as:

$$\underset{n\to \infty }{\lim }{a}_n=\infty$$

Analogously, one defines ${\lim }_{n\to \infty }{a}_n=-\infty$, and such sequence are also called improperly convergent. The values $+\infty ,-\infty$ are then referred to as improper limits.

Examples

$(2n+1{)}_{n\ge 0}$

$$(2n+1{)}_{n\ge 0}=(1,3,5,7,\dots )$$

We assume that ${\lim }_{n\to \infty }2n+1=\infty$. Let $K>0$ be given. Find a $N(K)$ such that for all $n>N(K)$ we have $2n+1>K$.

Trivially, we can find a $K$ such that $2n+1>K$ for all $n\ge 0$. However, this can be proven easily:

$$\begin{array}{rl}& 2n+1>K\\ ⟺& n>\frac{K-1}{2}\\ ⟹& N(K)=\lfloor \frac{K-1}{2}\rfloor \\ ⟹& \forall n>N(K):2n+1>K\\ ⟹& \underset{n\to \infty }{\lim }2n+1=\infty \end{array}$$