# 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 . ${\ displaystyle \ mathbb {R}}$