Definitions for divergent series
di·ver·gent se·ries

WiktionaryRate this definition:1.0 / 1 vote

1. divergent seriesnoun

   An infinite series whose partial sums are divergent

WikipediaRate this definition:0.0 / 0 votes

1. Divergent series

   In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. A counterexample is the harmonic series 1 + 1 2 + 1 3 + 1 4 + 1 5 + ⋯ = ∑ n = 1 ∞ 1 n . {\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots =\sum _{n=1}^{\infty }{\frac {1}{n}}.} The divergence of the harmonic series was proven by the medieval mathematician Nicole Oresme. In specialized mathematical contexts, values can be objectively assigned to certain series whose sequences of partial sums diverge, in order to make meaning of the divergence of the series. A summability method or summation method is a partial function from the set of series to values. For example, Cesàro summation assigns Grandi's divergent series 1 − 1 + 1 − 1 + ⋯ {\displaystyle 1-1+1-1+\cdots } the value 1/2. Cesàro summation is an averaging method, in that it relies on the arithmetic mean of the sequence of partial sums. Other methods involve analytic continuations of related series. In physics, there are a wide variety of summability methods; these are discussed in greater detail in the article on regularization.

WikidataRate this definition:0.0 / 0 votes

1. Divergent series

   In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a limit. If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. The simplest counterexample is the harmonic series The divergence of the harmonic series was proven by the medieval mathematician Nicole Oresme. In specialized mathematical contexts, values can be usefully assigned to certain series whose sequence of partial sums diverges. A summability method or summation method is a partial function from the set of sequences of partial sums of series to values. For example, Cesàro summation assigns Grandi's divergent series the value ¹/2. Cesàro summation is an averaging method, in that it relies on the arithmetic mean of the sequence of partial sums. Other methods involve analytic continuations of related series. In physics, there are a wide variety of summability methods; these are discussed in greater detail in the article on regularization.

How to pronounce divergent series?

1. Alex

   US English

   David

   US English

   Mark

   US English

   Daniel

   British

   Libby

   British

   Mia

   British

   Karen

   Australian

   Hayley

   Australian

   Natasha

   Australian

   Veena

   Indian

   Priya

   Indian

   Neerja

   Indian

   Zira

   US English

   Oliver

   British

   Wendy

   British

   Fred

   US English

   Tessa

   South African

How to say divergent series in sign language?

Numerology

1. Chaldean Numerology

   The numerical value of divergent series in Chaldean Numerology is: 9

2. Pythagorean Numerology

   The numerical value of divergent series in Pythagorean Numerology is: 8

References

1. ^ Wiktionary
   https://en.wiktionary.org/wiki/Divergent_Series
2. ^ Wikipedia
   https://en.wikipedia.org/wiki/Divergent_Series
3. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=divergent series