Zero sequence

In mathematics , a zero sequence is a sequence (mostly of real numbers ) that converges (approaches) towards 0 . Each convergent sequence can be represented as the sum of a constant number (namely its limit value) and a zero sequence.

For example, the sequence is a zero sequence of real numbers. ${\displaystyle (2^{-n})_{n\in \mathbb {N} }}$

definition

Be the field of real or complex numbers. A sequence is called a null sequence if ${\displaystyle \mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \}}$ ${\displaystyle (a_{n})_{n\in \mathbb {N} }\subset \mathbb {K} }$

${\displaystyle \lim \limits _{n\to \infty }a_{n}=0}$

applies. The set of all zero sequences forms the sequence space ${\displaystyle c_{0}}$ , which becomes   a Banach space with the supremum norm   . ${\displaystyle \textstyle \left\|(a_{n})_{n\in \mathbb {N} }\right\|_{\infty }:=\sup _{n\in \mathbb {N} }|a_{n}|}$

Examples

Examples of null sequences are: ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$

• ${\displaystyle a_{n}\,=\,0}$,

• ${\displaystyle a_{n}\,=\,{\frac {1}{n}}}$,

• ${\displaystyle a_{n}\,=\,{\frac {1}{n^{2}}}}$,

• ${\displaystyle a_{n}\,=\,(-1)^{n}{\frac {1}{n}}}$,

• ${\displaystyle a_{n}\,=\,(-0{,}5)^{n}}$,

• ${\displaystyle a_{n}\,=\,{\sqrt[{n}]{5}}-1}$.

generalization

Let be a metrizable topological group , i. H. a group that is equipped with a metric in such a way that the group connection and the formation of the inverse are continuous (e.g. the additive group in a weighted field or normalized vector space ). ${\displaystyle (G,+,d)}$

A sequence in is called a zero sequence if and only if it converges to the neutral element. ${\displaystyle G}$

The property of a result to be null sequence, of course, depends on the metric from: The above mentioned as an example result in a null sequence with respect to the usual amount of metric, but it diverges even with respect to the 2-adic amount to . ${\displaystyle a_{n}=(-0{,}5)^{n}}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Q} }$

A sequence in a standardized vector space is a null sequence with respect to the metric induced by the norm if and only if the sequence of the norms is a null sequence in . ${\displaystyle \mathbb {R} }$

See also

• Zero sequence criterion

swell

• Dirk Werner : Functional Analysis. Sixth, corrected edition. Springer-Verlag, Berlin 2007, ISBN 978-3-540-72533-6 , page 8

Web links

Wiktionary: Null sequence  - explanations of meanings, word origins, synonyms, translations

• Zero sequences on www.mathematik.net