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Some authors directly identify a series with its sequence of partial sums. [9] [11] Either the sequence of partial sums or the sequence of terms completely characterizes the series, and the sequence of terms can be recovered from the sequence of partial sums by taking the differences between consecutive elements, =.
For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common difference of 2. If the initial term of an arithmetic progression is a 1 {\displaystyle a_{1}} and the common difference of successive members is d {\displaystyle d} , then the n {\displaystyle n} -th term of the sequence ( a n {\displaystyle a_{n ...
The following is a selection from the large body of literature on absolutely/completely monotonic functions/sequences. René L. Schilling, Renming Song and Zoran Vondraček (2010). Bernstein Functions Theory and Applications. De Gruyter. pp. 1– 10. ISBN 978-3-11-021530-4. (Chapter 1 Laplace transforms and completely monotone functions)
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. [1] Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by Jean Leray , they have become an important research tool, particularly in homotopy theory.
In mathematics, a sequence is a list of objects (or events) which have been ordered in a sequential fashion; such that each member either comes before, or after, every other member. More formally, a sequence is a function with a domain equal to the set of positive integers. A series is a sum of a sequence of terms. That is, a series is a list ...
In mathematics, the Riemann series theorem, also called the Riemann rearrangement theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, and rearranged such that the new series diverges.
Now define sequences of rationals (u n) and (l n) as follows: Set u 0 = U and l 0 = L. For each n consider the number m n = (u n + l n)/2. If m n is an upper bound for S, set u n+1 = m n and l n+1 = l n. Otherwise set l n+1 = m n and u n+1 = u n. This defines two Cauchy sequences of rationals, and so the real numbers l = (l n) and u = (u n).