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Convergence Tests
In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series \sum_^\infty a_n. List of tests Limit of the summand If the limit of the summand is undefined or nonzero, that is \lim_a_n \ne 0, then the series must diverge. In this sense, the partial sums are Cauchy only if this limit exists and is equal to zero. The test is inconclusive if the limit of the summand is zero. This is also known as the nth-term test, test for divergence, or the divergence test. Ratio test This is also known as d'Alembert's criterion. : Consider two limits \ell=\liminf_\left, \frac\ and L=\limsup_\left, \frac\. If \ell>1, the series diverges. If L 1. The case of p = 1, k = 1 yields the harmonic series, which diverges. The case of p = 2, k = 1 is the Basel problem and the series converges to \frac. In general, for p > 1, k = 1, the series is equal to the Riemann zeta funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Comparison Test
In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing whether an infinite series or an improper integral converges or diverges by comparing the series or integral to one whose convergence properties are known. For series In calculus, the comparison test for series typically consists of a pair of statements about infinite series with non-negative ( real-valued) terms: * If the infinite series \sum b_n converges and 0 \le a_n \le b_n for all sufficiently large ''n'' (that is, for all n>N for some fixed value ''N''), then the infinite series \sum a_n also converges. * If the infinite series \sum b_n diverges and 0 \le b_n \le a_n for all sufficiently large ''n'', then the infinite series \sum a_n also diverges. Note that the series having larger terms is sometimes said to ''dominate'' (or ''eventually dominate'') the series with smaller terms. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotonic Function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus and analysis In calculus, a function f defined on a subset of the real numbers with real values is called ''monotonic'' if it is either entirely non-decreasing, or entirely non-increasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease. A function is termed ''monotonically increasing'' (also ''increasing'' or ''non-decreasing'') if for all x and y such that x \leq y one has f\!\left(x\right) \leq f\!\left(y\right), so f preserves the order (see Figure 1). Likewise, a function is called ''monotonically decreasing'' (also ''decreasing'' or ''non-increasing'') if, whenever x \leq y, then f\!\left(x\right) \geq f\!\left(y\right), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stolz–Cesàro Theorem
In mathematics, the Stolz–Cesàro theorem is a criterion for proving the convergence of a sequence. It is named after mathematicians Otto Stolz and Ernesto Cesàro, who stated and proved it for the first time. The Stolz–Cesàro theorem can be viewed as a generalization of the Cesàro mean, but also as a l'Hôpital's rule for sequences. Statement of the theorem for the case Let (a_n)_ and (b_n)_ be two sequences of real numbers. Assume that (b_n)_ is a strictly monotone and divergent sequence (i.e. strictly increasing and approaching + \infty , or strictly decreasing and approaching - \infty ) and the following limit exists: : \lim_ \frac=l.\ Then, the limit : \lim_ \frac=l.\ Statement of the theorem for the case Let (a_n)_ and (b_n)_ be two sequences of real numbers. Assume now that (a_n)\to 0 and (b_n)\to 0 while (b_n)_ is strictly decreasing. If : \lim_ \frac=l,\ then : \lim_ \frac=l.\ Proofs Proof of the theorem for the case Case 1: suppose (b_n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy's Convergence Test
The Cauchy convergence test is a method used to test infinite series for convergence. It relies on bounding sums of terms in the series. This convergence criterion is named after Augustin-Louis Cauchy who published it in his textbook '' Cours d'Analyse'' 1821. Statement A series \sum_^\infty a_i is convergent if and only if for every \varepsilon>0 there is a natural number N such that :, a_+a_+\cdots+a_, N and all p \geq 1. Explanation The test works because the space \R of real numbers and the space \C of complex numbers (with the metric given by the absolute value) are both complete. From here, the series is convergent if and only if the partial sums : s_n := \sum_^n a_i are a Cauchy sequence. Cauchy's convergence test can only be used in complete metric spaces (such as \R and \C), which are spaces where all Cauchy sequences converge. This is because we need only show that its elements become arbitrarily close to each other after a finite progression in the sequence to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers, sometimes called "the reals", is traditionally denoted by a bold , often using blackboard bold, . The adjective ''real'', used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of . The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter "M" first and "Y" last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be '' finite'', as in these examples, or '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet's Test
In mathematics, Dirichlet's test is a method of testing for the convergence of a series that is especially useful for proving conditional convergence. It is named after its author Peter Gustav Lejeune Dirichlet, and was published posthumously in the '' Journal de Mathématiques Pures et Appliquées'' in 1862. Statement The test states that if (a_n) is a monotonic sequence of real numbers with \lim_ a_n = 0 and (b_n) is a sequence of real numbers or complex numbers with bounded partial sums, then the series :\sum_^ a_n b_n converges. Proof Let S_n = \sum_^n a_k b_k and B_n = \sum_^n b_k. From summation by parts, we have that S_n = a_ B_n + \sum_^ B_k (a_k - a_). Since the magnitudes of the partial sums B_n are bounded by some ''M'' and a_n \to 0 as n\to\infty, the first of these terms approaches zero: , a_ B_n, \leq , a_ M, \to 0 as n\to\infty. Furthermore, for each ''k'', , B_k (a_k - a_), \leq M, a_k - a_, . Since (a_n) is monotone, it is either decreasing or in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Series Test
In mathematical analysis, the alternating series test proves that an alternating series is convergent when its terms decrease monotonically in absolute value and approach zero in the limit. The test was devised by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion. The test is only sufficient, not necessary, so some convergent alternating series may fail the first part of the test. For a generalization, see Dirichlet's test. History Leibniz discussed the criterion in his unpublished ''De quadratura arithmetica'' of 1676 and shared his result with Jakob Hermann in June 1705 and with Johann Bernoulli in October, 1713. It was only formally published in 1993. Formal statement Alternating series test A series of the form \sum_^\infty (-1)^ a_n = a_0-a_1 + a_2 - a_3 + \cdots where either all ''a''''n'' are positive or all ''a''''n'' are negative, is called an alternating series. The alternating series test guarantees that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Convergence
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said to converge absolutely if \textstyle\sum_^\infty \left, a_n\ = L for some real number \textstyle L. Similarly, an improper integral of a function, \textstyle\int_0^\infty f(x)\,dx, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if \textstyle\int_0^\infty , f(x), dx = L. A convergent series that is not absolutely convergent is called conditionally convergent. Absolute convergence is important for the study of infinite series, because its definition guarantees that a series will have some "nice" behaviors of finite sums that not all convergent series possess. For instance, rearrangements do not change the value of the sum, which is not necessarily true for conditionally converge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abel's Test
In mathematics, Abel's test (also known as Abel's criterion) is a method of testing for the convergence of an infinite series. The test is named after mathematician Niels Henrik Abel, who proved it in 1826. There are two slightly different versions of Abel's test – one is used with series of real numbers, and the other is used with power series in complex analysis. Abel's uniform convergence test is a criterion for the uniform convergence of a series of functions dependent on parameters. Abel's test in real analysis Suppose the following statements are true: # \sum a_n is a convergent series, # b_n is a monotone sequence, and # b_n is bounded. Then \sum a_nb_n is also convergent. It is important to understand that this test is mainly pertinent and useful in the context of non absolutely convergent series \sum a_n. For absolutely convergent series, this theorem, albeit true, is almost self evident. This theorem can be proved directly using summation by parts. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
