Cauchy’s integral test
E825422
Cauchy’s integral test is a convergence criterion in mathematical analysis that determines whether an infinite series converges by relating it to the behavior of a corresponding improper integral.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
convergence test
ⓘ
mathematical theorem ⓘ result in real analysis ⓘ |
| appearsIn |
textbooks on advanced calculus
ⓘ
textbooks on real analysis ⓘ |
| appliesTo |
infinite series
ⓘ
series of real numbers ⓘ |
| assumes |
f is continuous on [1,∞)
ⓘ
f is decreasing on [1,∞) ⓘ f is nonnegative on [1,∞) ⓘ monotone decreasing terms ⓘ positive terms ⓘ terms given by a function f(n) ⓘ |
| category |
theorem about improper integrals
ⓘ
theorem about infinite series ⓘ |
| compares |
∑ f(n)
ⓘ
∫₁^∞ f(x) dx ⓘ |
| concludes |
∑ 1/n^p converges for p>1 via integral of x^{-p}
ⓘ
∑ 1/n^p diverges for 0<p≤1 via integral of x^{-p} ⓘ |
| criterionType | convergence criterion ⓘ |
| field |
mathematical analysis
ⓘ
real analysis NERFINISHED ⓘ |
| formalizes | link between discrete sums and continuous integrals ⓘ |
| hasPrerequisite |
basic real analysis
ⓘ
knowledge of improper integrals ⓘ knowledge of infinite series ⓘ |
| historicalPeriod | 19th-century mathematics ⓘ |
| holdsFor | series with terms f(n)=1/n^p where p>0 ⓘ |
| implies |
if ∫₁^∞ f(x) dx is finite then ∑ f(n) is convergent
ⓘ
if ∫₁^∞ f(x) dx is infinite then ∑ f(n) is divergent ⓘ |
| logicalForm | if and only if statement between series and integral convergence under hypotheses ⓘ |
| namedAfter | Augustin-Louis Cauchy NERFINISHED ⓘ |
| relatedTo |
Cauchy condensation test
NERFINISHED
ⓘ
comparison test ⓘ integral comparison methods ⓘ p-series test ⓘ |
| relates |
improper integrals
ⓘ
series convergence ⓘ |
| requires | comparison between discrete sum and continuous integral ⓘ |
| states |
the series ∑ f(n) converges if the improper integral ∫₁^∞ f(x) dx converges
ⓘ
the series ∑ f(n) diverges if the improper integral ∫₁^∞ f(x) dx diverges ⓘ |
| usedFor |
testing convergence of p-series
ⓘ
testing convergence of series with algebraic terms ⓘ testing convergence of series with logarithmic factors ⓘ |
| usedIn |
calculus courses
ⓘ
problems involving asymptotic behavior of series ⓘ undergraduate analysis courses ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Augustin-Louis Cauchy