Convergent Sequence

analysis

Definition

Convergent Sequence

A sequence $(a_n{)}_{n\ge 0}$ is called convergent if it has a limit.

Example: The limit of the below sequence is:

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

which can also be written as:

$$a_n⟶a$$

Improperly Convergent

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.

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