Dirichlet test
E466254
The Dirichlet test is a criterion in mathematical analysis that provides sufficient conditions for the convergence of certain infinite series, particularly those involving oscillatory terms.
Statements (33)
| Predicate | Object |
|---|---|
| instanceOf |
concept in mathematical analysis
ⓘ
convergence test ⓘ mathematical criterion ⓘ |
| appliesTo |
oscillatory series
ⓘ
series of complex numbers ⓘ series of real numbers ⓘ |
| assumes |
b_n is monotone
ⓘ
limit of b_n equals 0 ⓘ sequence of partial sums of a_n is bounded ⓘ |
| category | tests for convergence of series ⓘ |
| concludes | series sum a_n b_n converges ⓘ |
| conditionOn |
partial sums of one factor are uniformly bounded
ⓘ
the other factor forms a monotone sequence tending to 0 ⓘ |
| contrastsWith | absolute convergence tests ⓘ |
| field |
infinite series
ⓘ
mathematical analysis ⓘ |
| generalizationOf | some special cases of alternating series convergence ⓘ |
| hasVariant | Dirichlet test for improper integrals NERFINISHED ⓘ |
| historicalPeriod | 19th-century mathematics ⓘ |
| implies | conditional convergence when absolute convergence fails in some examples ⓘ |
| namedAfter | Peter Gustav Lejeune Dirichlet NERFINISHED ⓘ |
| namedEntity | true ⓘ |
| provides | sufficient condition for convergence ⓘ |
| relatedTo |
Abel test
ⓘ
Fourier series NERFINISHED ⓘ alternating series test ⓘ |
| strongerThan | alternating series test in some cases ⓘ |
| typicalForm | sum a_n b_n ⓘ |
| usedFor |
proving conditional convergence of series
ⓘ
proving convergence of Fourier series at a point ⓘ |
| usedIn |
analysis of Fourier coefficients
ⓘ
proofs of convergence for series with sine and cosine terms ⓘ theory of trigonometric series ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.