Tannery's Theorem
In mathematical analysis, Tannery's theorem gives sufficient conditions for the interchanging of the limit and infinite summation operations. It is named after Jules Tannery. Statement Let S_n = \sum_^\infty a_k(n) and suppose that \lim_ a_k(n) = b_k . If , a_k(n), \le M_k and \sum_^\infty M_k < \infty , then $\backslash lim\_\; S\_n\; =\; \backslash sum\_^\; b\_k$.

Proofs

Tannery's theorem follows directly from Lebesgue's applied to the

Interchange Of Limiting Operations
In mathematics, the study of interchange of limiting operations is one of the major concerns of mathematical analysis, in that two given limiting operations, say ''L'' and ''M'', cannot be ''assumed'' to give the same result when applied in either order. One of the historical sources for this theory is the study of trigonometric series. Formulation In symbols, the assumption :''LM'' = ''ML'', where the left-hand side means that ''M'' is applied first, then ''L'', and ''vice versa'' on the right-hand side, is not a valid equation between mathematical operators, under all circumstances and for all operands. An algebraist would say that the operations do not commute. The approach taken in analysis is somewhat different. Conclusions that assume limiting operations do 'commute' are called ''formal''. The analyst tries to delineate conditions under which such conclusions are valid; in other words mathematical rigour is established by the specification of some set of sufficient cond ...
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Jules Tannery
Jules Tannery (24 March 1848 – 11 December 1910) was a French mathematician, who notably studied under Charles Hermite and was the PhD advisor of Jacques Hadamard. Tannery's theorem on interchange of limits and series is named after him. He was a brother of the mathematician and historian of science Paul Tannery. Under Hermite, he received a doctorate in 1874 for his thesis ''Propriétés des intégrales des équations différentielles linéaires à coefficients variables.'' Tannery was an advocate for mathematics education, particular as a means to train children in logical consequence through synthetic geometry and mathematical proofs. Tannery discovered a surface of the fourth order of which all the geodesic lines are algebraic. He was not an inventor, however, but essentially a critic and methodologist. He once remarked, "Mathematicians are so used to their symbols and have so much fun playing with them, that it is sometimes necessary to take their toys away from them in ...
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Dominated Convergence Theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary theoretical advantages of Lebesgue integration over Riemann integration. In addition to its frequent appearance in mathematical analysis and partial differential equations, it is widely used in probability theory, since it gives a sufficient condition for the convergence of expected values of random variables. Statement Lebesgue's dominated convergence theorem. Let (f_n) be a sequence of complex-valued measurable functions on a measure space . Suppose that the sequence converges pointwise to a function f and is dominated by some integrable function g in the sense that : , f_n(x), \le g(x) for all numbers ''n'' in the index set of the sequence and all points x\in S. Then ''f'' is integrable (in the Lebesgue sense) and : \lim_ \int_ ...
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Sequence Space
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field ''K'' of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in ''K'', and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space. The most important sequence spaces in analysis are the spaces, consisting of the -power summable sequences, with the ''p''-norm. These are special cases of L''p'' spaces for the counting measure on the set of natural numbers. Other important classes of sequences like ...
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Characterizations Of The Exponential Function
In mathematics, the exponential function can be characterized in many ways. The following characterizations (definitions) are most common. This article discusses why each characterization makes sense, and why the characterizations are independent of and equivalent to each other. As a special case of these considerations, it will be demonstrated that the three most common definitions given for the mathematical constant ''e'' are equivalent to each other. Characterizations The six most common definitions of the exponential function for real are: # Define by the limit e^x = \lim_ \left(1+\frac x n \right)^n. # Define as the value of the infinite series e^x = \sum_^\infty = 1 + x + \frac + \frac + \frac + \cdots (Here denotes the factorial of . One proof that is irrational uses a special case of this formula.) # Define to be the unique number such that \int_1^y \frac = x. This is as the inverse of the natural logarithm function, which is defined by this integral. # Define ...
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Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were i ...
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