Convergent Sequence

analysis

Definition

Convergent Sequence

A sequence is called convergent if it has a limit.

Example: The limit of the below sequence is:

which can also be written as:

Improperly Convergent

Definition

Uneigentlich Konvergente Folge

Improperly Convergent Sequence

A sequence whose elements become arbitrarily large with:

and can be denoted as as:

Analogously, one defines , and such sequence are also called improperly convergent. The values are then referred to as improper limits.

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Limitedness

All convergent sequences are bounded.

Let a sequence with and . The sequence elements with are . Let (or in the case ). Then, in particular, , and hence all sequence elements lie in . The interval boundaries and then form the lower and upper bounds of .

Cauchy Sequence

Definition

Cauchy Sequence

A sequence is called cauchy sequence, named after Augustin-Louis Cauchy, if for all there exists an , such that:

Cauchy sequences are convergent and all convergent sequences are Cauchy sequences.

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Examples

Example: