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Cauchy's mean value theorem, also known as the extended mean value theorem, is a generalization of the mean value theorem. [ 6 ] [ 7 ] It states: if the functions f {\displaystyle f} and g {\displaystyle g} are both continuous on the closed interval [ a , b ] {\displaystyle [a,b]} and differentiable on the open interval ( a , b ) {\displaystyle ...
This is also known as the nth root test or Cauchy's criterion. Let = | |, where denotes the limit superior (possibly ; if the limit exists it is the same value). If r < 1, then the series converges absolutely.
The Cauchy condensation test follows from the stronger estimate, = = = (), which should be understood as an inequality of extended real numbers.The essential thrust of a proof follows, patterned after Oresme's proof of the divergence of the harmonic series.
Several theorems are named after Augustin-Louis Cauchy. Cauchy theorem may mean: Cauchy's integral theorem in complex analysis, also Cauchy's integral formula; Cauchy's mean value theorem in real analysis, an extended form of the mean value theorem; Cauchy's theorem (group theory) Cauchy's theorem (geometry) on rigidity of convex polytopes
As an application of the above estimate, we can obtain the Stieltjes–Vitali theorem, [5] which says that a sequence of holomorphic functions on an open subset that is bounded on each compact subset has a subsequence converging on each compact subset (necessarily to a holomorphic function since the limit satisfies the Cauchy–Riemann equations).
A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions that are given on a hypersurface in the domain. [1] A Cauchy problem can be an initial value problem or a boundary value problem (for this case see also Cauchy boundary condition). It is named after Augustin-Louis Cauchy.