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A series is convergent (or converges) if and only if the sequence (,,, … ) {\displaystyle (S_{1},S_{2},S_{3},\dots )} of its partial sums tends to a limit ; that means that, when adding one a k {\displaystyle a_{k}} after the other in the order given by the indices , one gets partial sums that become closer and closer to a given number.
The plot of a convergent sequence {a n} is shown in blue. Here, one can see that the sequence is converging to the limit 0 as n increases. In the real numbers, a number is the limit of the sequence (), if the numbers in the sequence become closer and closer to , and not to any other number.
The theorem states that each infinite bounded sequence in has a convergent subsequence. [1] An equivalent formulation is that a subset of R n {\displaystyle \mathbb {R} ^{n}} is sequentially compact if and only if it is closed and bounded . [ 2 ]
Every bounded-above monotonically nondecreasing sequence of real numbers is convergent in the real numbers because the supremum exists and is a real number. The proposition does not apply to rational numbers because the supremum of a sequence of rational numbers may be irrational.
Then divide the sequence into an initial segment, and a tail of small terms: given any ε > 0 we can take N large enough to make the initial segment of terms up to c N average to at most ε/2, while each term in the tail is bounded by ε/2 so that the average is also necessarily bounded. The name derives from Abel's theorem on power series.
A function from a metric space to another metric space is continuous exactly when it takes convergent sequences to convergent sequences. A metric space is a connected space if and only if, whenever the space is partitioned into two sets, one of the two sets contains a sequence converging to a point in the other set.
Given a real number r ≥ 1, we say that the sequence X n converges in the r-th mean (or in the L r-norm) towards the random variable X, if the r-th absolute moments (|X n | r ) and (|X| r ) of X n and X exist, and
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.