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In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the symbol (e.g., ). [1] If such a limit exists and is finite, the sequence is called convergent. [2]
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 value a random variable will take, rather than just the distribution.
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.
explicitly, for every Mackey convergent null sequence () = in (), the sequence (()) = is bounded. a sequence = = is said to be Mackey convergent to 0 if there exists a divergent sequence = = of positive real number such that the sequence () = is bounded; every sequence that is Mackey convergent to 0 necessarily converges to the origin (in the ...
Because () is bounded, this sequence has a lower bound and an upper bound . We take I 1 = [ s , S ] {\displaystyle I_{1}=[s,S]} as the first interval for the sequence of nested intervals. Then we split I 1 {\displaystyle I_{1}} at the mid into two equally sized subintervals.
A sequence that does not converge is divergent. Informally, a sequence has a limit if the elements of the sequence become closer and closer to some value (called the limit of the sequence), and they become and remain arbitrarily close to , meaning that given a real number greater than zero, all but a finite number of the elements of the ...
A series is said to be convergent if the sequence consisting of its partial sums, (), is convergent; otherwise it is divergent. The sum of a convergent series is defined as the number s = lim n → ∞ s n {\textstyle s=\lim _{n\to \infty }s_{n}} .
Remark 5 The stronger version of the dominated convergence theorem can be reformulated as: if a sequence of measurable complex functions is almost everywhere pointwise convergent to a function and almost everywhere bounded in absolute value by an integrable function then in the Banach space (,)