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Contrary to Fatou's lemma, this value is strictly less than the integral of the limit (0). As discussed in § Extensions and variations of Fatou's lemma below, the problem is that there is no uniform integrable bound on the sequence from below, while 0 is the uniform bound from above.
In multivariable calculus, an iterated limit is a limit of a sequence or a limit of a function in the form , = (,), (,) = ((,)),or other similar forms. An iterated limit is only defined for an expression whose value depends on at least two variables. To evaluate such a limit, one takes the limiting process as one of the two variables approaches some number, getting an expression whose value ...
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]
The equals sign, used to represent equality symbolically in an equation. In mathematics, equality is a relationship between two quantities or expressions, stating that they have the same value, or represent the same mathematical object.
The uniform limit has "nicer" properties than the pointwise limit. For example, the uniform limit of a sequence of continuous functions is continuous. Many different notions of convergence can be defined on function spaces. This is sometimes dependent on the regularity of the space. Prominent examples of function spaces with some notion of ...
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
List of limits; List of logarithmic identities; List of mathematical functions; List of mathematical identities; List of mathematical proofs; List of misnamed theorems; List of scientific laws; List of theories; Most of the results below come from pure mathematics, but some are from theoretical physics, economics, and other applied fields.
The example function (,) = (+) illustrates that the equality does not hold for every function. A theorem giving conditions on f, W, and Z which guarantee the saddle point property is called a minimax theorem.
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