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If r < 1, then the series converges absolutely. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge. The root test is stronger than the ratio test: whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely. [1]
In asymptotic analysis in general, one sequence () that converges to a limit is said to asymptotically converge to with a faster order of convergence than another sequence () that converges to in a shared metric space with distance metric | |, such as the real numbers or complex numbers with the ordinary absolute difference metrics, if
The continuous mapping theorem states that for a continuous function g, if the sequence {X n} converges in distribution to X, then {g(X n)} converges in distribution to g(X). Note however that convergence in distribution of {X n} to X and {Y n} to Y does in general not imply convergence in distribution of {X n + Y n} to X + Y or of {X n Y n} to XY.
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.
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 bounded continuous function. Now consider the function of a single variable g(x) := f(x, c).
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 of sequence.
If the integral of a function f is uniformly bounded over all intervals, and g is a non-negative monotonically decreasing function, then the integral of fg is a convergent improper integral. Notes [ edit ]
The test is as follows. Let {g n} be a uniformly bounded sequence of real-valued continuous functions on a set E such that g n+1 (x) ≤ g n (x) for all x ∈ E and positive integers n, and let {f n} be a sequence of real-valued functions such that the series Σf n (x) converges uniformly on E. Then Σf n (x)g n (x) converges uniformly on E.