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In mathematics, the limit comparison test (LCT) (in contrast with the related direct comparison test) is a method of testing for the convergence of an infinite series. Statement [ edit ]
create limits for F if whenever (L, φ) is a limit of GF there exists a unique cone (L′, φ′) to F such that G(L′, φ′) = (L, φ), and furthermore, this cone is a limit of F. reflect limits for F if each cone to F whose image under G is a limit of GF is already a limit of F. Dually, one can define creation and reflection of colimits.
In general, any infinite series is the limit of its partial sums. For example, an analytic function is the limit of its Taylor series, within its radius of convergence. = =. This is known as the harmonic series. [6]
For example, it is possible to construct a sequence of continuous functions which has a discontinuous pointwise limit. Another notion of convergence is uniform convergence . The uniform distance between two functions f , g : E → R {\displaystyle f,g:E\rightarrow \mathbb {R} } is the maximum difference between the two functions as the argument ...
Limit of a function (ε,_δ)-definition of limit, formal definition of the mathematical notion of limit; Limit of a sequence; One-sided limit, either of the two limits of a function as a specified point is approached from below or from above; Limit inferior and limit superior; Limit of a net; Limit point, in topological spaces; Limit (category ...
A reference to a standard or choice-free presentation of some mathematical object (e.g., canonical map, canonical form, or canonical ordering). The same term can also be used more informally to refer to something "standard" or "classic". For example, one might say that Euclid's proof is the "canonical proof" of the infinitude of primes.
Fulton–Hansen connectedness theorem (algebraic geometry) Fundamental theorem of algebra (complex analysis) Fundamental theorem of arbitrage-free pricing (financial mathematics) Fundamental theorem of arithmetic (number theory) Fundamental theorem of calculus ; Fundamental theorem of equivalence relations
In Riemannian geometry, it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry. [4] Rauch comparison theorem relates the sectional curvature of a Riemannian manifold to the rate at which its geodesics spread apart; Toponogov's theorem; Myers's theorem; Hessian ...
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