Cauchy condensation test
E239293
The Cauchy condensation test is a convergence criterion in mathematical analysis that determines whether an infinite series with positive, nonincreasing terms converges by comparing it to a related series formed by powers of two.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Cauchy condensation test canonical | 2 |
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
convergence test
ⓘ
mathematical criterion ⓘ theorem in real analysis ⓘ |
| appliesTo |
infinite series
ⓘ
series with nonincreasing terms ⓘ series with positive terms ⓘ |
| appliesWhen | terms decrease sufficiently regularly ⓘ |
| assumes | index set is the positive integers ⓘ |
| attributedTo | Augustin-Louis Cauchy ⓘ |
| category | series convergence test ⓘ |
| compares |
condensed series
ⓘ
original series ⓘ |
| conditionOnTerms |
terms must be nonincreasing
ⓘ
terms must be nonnegative ⓘ |
| convergenceCriterion | sum a_n converges iff sum 2^n a_{2^n} converges ⓘ |
| definesCondensedSeries | sum of 2^n a_{2^n} ⓘ |
| doesNotApplyTo | series with sign-changing terms without modification ⓘ |
| field |
mathematical analysis
ⓘ
real analysis ⓘ |
| helpsShow |
divergence of harmonic series
ⓘ
growth rate of partial sums for some series ⓘ |
| implication |
sum 1/n^p converges for p>1
ⓘ
sum 1/n^p diverges for p<=1 ⓘ |
| language | stated in terms of sequences and series ⓘ |
| logicalForm | biconditional between convergence of two series ⓘ |
| namedAfter | Augustin-Louis Cauchy ⓘ |
| originalSeriesForm | sum of a_n from n=1 to infinity ⓘ |
| proofTechnique |
comparison of grouped sums with condensed series
ⓘ
grouping terms in blocks of dyadic length ⓘ |
| relatedTo |
comparison test
ⓘ
integral test ⓘ p-series test ⓘ |
| requires | monotone decreasing sequence of terms ⓘ |
| resultType | necessary and sufficient condition for convergence ⓘ |
| termType | real nonnegative terms ⓘ |
| typicalExample | series sum 1/n^p ⓘ |
| usedBy |
mathematicians studying series
ⓘ
students of analysis ⓘ |
| usedFor |
analyzing series with slowly decreasing terms
ⓘ
testing convergence of series similar to p-series ⓘ |
| usedIn |
calculus courses
ⓘ
undergraduate real analysis courses ⓘ |
| usesTransformation |
dyadic subsequence of terms
ⓘ
powers of two in the index ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Augustin-Louis Cauchy