Assertion: series | /r/HasProperty | convergent
Facets:
value: absolutely
type: DEGREE
value: conversely
type: DEGREE
value: conditionally
type: MANNER
value: unconditionally
type: MANNER
Source sentences:
text: If f satisfies Dirichlet conditions, then for all x, we have that the series obtained by plugging x into the Fourier series is convergent, and is given by
text: A series is said to be semi-convergent (or conditionally convergent) if it is convergent but not absolutely convergent.
text: When this limit exists, one says that the series is convergent or summable, or that the sequence ( a 1 , a 2 , a 3 , … ) {\displaystyle (a_{1},a_{2},a_{3},\ldots )} is summable.
text: A series is said to be convergent if it converges to some limit or divergent when it does not.
text: A series ∑a n is said to converge or to be convergent when the sequence (s k ) of partial sums has a finite limit.
text: then the series ∑ a n is convergent.
text: Cauchy (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with him begins the discovery of effective criteria.
text: A series is said to be semi-convergent (or conditionally convergent) if it is convergent but not absolutely convergent.
text: Conversely, if every permutation of a series ∑a n converges, then the series is unconditionally convergent.
text: A series is convergent if the sequence of its partial sums ( S 1 , S 2 , S 3 , … ) {\displaystyle (S_{1},S_{2},S_{3},\dots )} tends to a limit; that means that the partial sums become closer and closer to a given number when the number of their terms increases.
text: If the series is convergent, the number ℓ {\displaystyle \ell } (necessarily unique) is called the sum of the series.
text: If r < 1, then the series is absolutely convergent.
text: That is, the series is absolutely and compactly convergent on the interior of the disc of convergence.