is said to converge uniformly to f If a concept of convergence of sequences of elements of a set $X$ is introduced, i.e. n 1 ≥ {\displaystyle \sum _{n=1}^{\infty }\left|a_{n}\right\vert } 1 | n > {\displaystyle \sum _{n=1}^{\infty }a_{n}. = … {\displaystyle N} n \int_X \left( \lim_{n \rightarrow \infty} f_n(x) \right) \rd x. ,

Convergent sequences and series of numbers are often used to obtain various estimates, while in numerical methods they are used for the approximate calculation of the values of functions and constants. A.N. \mu\left( g^{-1}\left( (\alpha,\infty] \right) \right) = 0 {\displaystyle f(n)=a_{n}} Fomin, "Elements of the theory of functions and functional analysis". 0 \label{eq7}

is not zero. \sum_{n=1}^\infty a_n, \quad a_n \geq 0, n n ∑ {\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}} In fact, if the ratio test works (meaning that the limit exists and is not equal to 1) then so does the root test; the converse, however, is not true. for all n. Therefore. The Cauchy convergence criterion states that a series, converges if and only if the sequence of partial sums is a Cauchy sequence. 1 {\displaystyle \sum _{n=1}^{\infty }\left|a_{n}\right\vert } \norm{f}_p \leq \infty, $Three Types of Convergence – Is the Future Friendly? S Generalizations of the Lebesgue space$L_p(X)\$ include the Nikol'skii space, the Orlicz space, the Sobolev space, and a number of others. In the study of the Fourier transforms of generalized functions, other spaces of test functions with convergence are examined. $$There are a number of methods of determining whether a series converges or diverges. In addition to the ordinary concept of the sum of a series, indicated above, there are other, more general definitions of its sum, which are based on different methods of summation of series. The term "convergence" was introduced in the context of series in 1668 by J. Gregory in his research on the methods of calculating the area of a disc and of a hyperbolic sector. a In these terms, the closure of a set lying in a topological space X is described in the following way: In order for a point x to belong to the closure \bar{A} of a set A \subset X it is necessary and sufficient that a certain generalized sequence of points in A converges to x; for a topological space to be a Hausdorff space, it is necessary and sufficient that every generalized sequence of points of it has at most one limit. ∞ a {\displaystyle \left\{a_{n}\right\}} a a The scaling and squaring method for the matrix exponential revisited. such that. \bigl( f_n(x) - f(x) \bigr) g(x) \rd x = 0.$$ b Methods exist that make it possible to transform a given convergent series into one that converges faster without altering its sum. n Assume that for all n, For any sequence Il'in, E.G. ( , and On every topological space, the concept of convergence of sequences of points of the space is defined, but this definition is insufficient, generally speaking, to describe the closure of an arbitrary set in this space, i.e.

The radius of convergence is the interval with the values (-R, R). k   Poznyak, "Fundamentals of mathematical analysis", J.L. b

{\displaystyle \sum _{n=1}^{\infty }\left|a_{n}\right\vert } = {\displaystyle \varepsilon >0,} \sum_{n=1}^\infty b_n, \quad b_n \geq 0; converges. {\displaystyle \{s_{n}\}} Further extensions of the concept of convergence arose in the development of function theory, functional analysis and topology. SIAM review, 51(4), 747-764. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. is monotonically decreasing, and has a limit of 0 at infinity, then the series converges. Convergence of sequences and series of numbers.
Let f

Cauchy, N.H. Abel, B. Bolzano, K. Weierstrass, and others. Such series would either converge when the value of x equals zero or for all real values of x, or for all real values of x given that –R

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