Search results
Results From The WOW.Com Content Network
Proof: We will prove this statement using the portmanteau lemma, part A. First we want to show that (X n, c) converges in distribution to (X, c). By the portmanteau lemma this will be true if we can show that E[f(X n, c)] → E[f(X, c)] for any bounded continuous function f(x, y). So let f be such arbitrary
Convergence proof techniques are canonical patterns of mathematical proofs that sequences or functions converge to a finite limit when the argument tends to infinity. There are many types of sequences and modes of convergence , and different proof techniques may be more appropriate than others for proving each type of convergence of each type ...
Abel's uniform convergence test is a criterion for the uniform convergence of a series of functions or an improper integration of functions dependent on parameters. It is related to Abel's test for the convergence of an ordinary series of real numbers, and the proof relies on the same technique of summation by parts. The test is as follows.
The theorem is still valid in a Banach algebra (see first line of the following proof). It is not sufficient for both series to be convergent; if both sequences are conditionally convergent, the Cauchy product does not have to converge towards the product of the two series, as the following example shows:
In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely.It applies to series whose terms are bounded functions with real or complex values, and is analogous to the comparison test for determining the convergence of series of real or complex numbers.
In mathematics, Dirichlet's test is a method of testing for the convergence of a series that is especially useful for proving conditional convergence. It is named after its author Peter Gustav Lejeune Dirichlet, and was published posthumously in the Journal de Mathématiques Pures et Appliquées in 1862. [1]
The above examples involving the harmonic series raise the question of whether there are monotone sequences such that f(n) decreases to 0 faster than 1/n but slower than 1/n 1+ε in the sense that lim n → ∞ f ( n ) 1 / n = 0 and lim n → ∞ f ( n ) 1 / n 1 + ε = ∞ {\displaystyle \lim _{n\to \infty }{\frac {f(n)}{1/n}}=0\quad {\text{and ...
The different notions of convergence capture different properties about the sequence, with some notions of convergence being stronger than others. For example, convergence in distribution tells us about the limit distribution of a sequence of random variables. This is a weaker notion than convergence in probability, which tells us about the ...