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Alternating Series
In mathematics, an alternating series is an infinite series of the form \sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n with for all . The signs of the general terms alternate between positive and negative. Like any series, an alternating series converges if and only if the associated sequence of partial sums converges. Examples The geometric series 1/2 − 1/4 %2B 1/8 − 1/16 %2B %E2%8B%AF sums to 1/3. The alternating harmonic series has a finite sum but the harmonic series does not. The Mercator series provides an analytic expression of the natural logarithm: \sum_^\infty \frac x^n \;=\; \ln (1+x). The functions sine and cosine used in trigonometry can be defined as alternating series in calculus even though they are introduced in elementary algebra as the ratio of sides of a right triangle. In fact, \sin x = \sum_^\infty (-1)^n \frac, and \cos x = \sum_^\infty (-1)^n \frac . When the alternating factor is removed from these series one obtains the hyperbolic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...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 and only if it is either entirely non-increasing, or entirely non-decreasing. 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 called ''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\r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Series Acceleration
In mathematics, series acceleration is one of a collection of sequence transformations for improving the rate of convergence of a series. Techniques for series acceleration are often applied in numerical analysis, where they are used to improve the speed of numerical integration. Series acceleration techniques may also be used, for example, to obtain a variety of identities on special functions. Thus, the Euler transform applied to the hypergeometric series gives some of the classic, well-known hypergeometric series identities. Definition Given a sequence :S=\_ having a limit :\lim_ s_n = \ell, an accelerated series is a second sequence :S'=\_ which converges faster to \ell than the original sequence, in the sense that :\lim_ \frac = 0. If the original sequence is divergent, the sequence transformation acts as an extrapolation method to the antilimit \ell. The mappings from the original to the transformed series may be linear (as defined in the article sequence tr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Series Theorem
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, or diverges. This implies that a series of real numbers is absolutely convergent if and only if it is unconditionally convergent. As an example, the series 1 − 1 + 1/2 − 1/2 + 1/3 − 1/3 + ⋯ converges to 0 (for a sufficiently large number of terms, the partial sum gets arbitrarily near to 0); but replacing all terms with their absolute values gives 1 + 1 + 1/2 + 1/2 + 1/3 + 1/3 + ⋯, which sums to infinity. Thus the original series is conditionally convergent, and can be rearranged (by taking the first two positive terms followed by the first negative term, followed by the next two positive terms and then the next negative term, etc.) t ... [...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. Absolute convergence is important for the study of infinite series because its definition is strong enough to have properties of finite sums that not all convergent series possess - a convergent series that is not absolutely convergent is called conditionally convergent, while absolutely convergent series behave "nicely". For instance, rearrangements do not change the value of the sum. This is not true for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Series (mathematics)
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical. This paradox was resolved using the concept of a limit during the 17th century. Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Series
In mathematics, an alternating series is an infinite series of the form \sum_^\infty (-1)^n a_n or \sum_^\infty (-1)^ a_n with for all . The signs of the general terms alternate between positive and negative. Like any series, an alternating series converges if and only if the associated sequence of partial sums converges. Examples The geometric series 1/2 − 1/4 %2B 1/8 − 1/16 %2B %E2%8B%AF sums to 1/3. The alternating harmonic series has a finite sum but the harmonic series does not. The Mercator series provides an analytic expression of the natural logarithm: \sum_^\infty \frac x^n \;=\; \ln (1+x). The functions sine and cosine used in trigonometry can be defined as alternating series in calculus even though they are introduced in elementary algebra as the ratio of sides of a right triangle. In fact, \sin x = \sum_^\infty (-1)^n \frac, and \cos x = \sum_^\infty (-1)^n \frac . When the alternating factor is removed from these series one obtains the hyperbolic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Convergence
In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely. Definition More precisely, a series of real numbers \sum_^\infty a_n is said to converge conditionally if \lim_\,\sum_^m a_n exists (as a finite real number, i.e. not \infty or -\infty), but \sum_^\infty \left, a_n\ = \infty. A classic example is the alternating harmonic series given by 1 - + - + - \cdots =\sum\limits_^\infty , which converges to \ln (2), but is not absolutely convergent (see Harmonic series). Bernhard Riemann proved that a conditionally convergent series may be rearranged to converge to any value at all, including ∞ or −∞; see ''Riemann series theorem''. The Lévy–Steinitz theorem identifies the set of values to which a series of terms in R''n'' can converge. A typical conditionally convergent integral is that on the non-negative real axis of \sin (x^2) (see Fresnel integral). See also *Absolute converg ... [...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 the convergence or divergence of an infinite series or an improper integral. In both cases, the test works by comparing the given 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 d ... [...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. Absolute convergence is important for the study of infinite series because its definition is strong enough to have properties of finite sums that not all convergent series possess - a convergent series that is not absolutely convergent is called conditionally convergent, while absolutely convergent series behave "nicely". For instance, rearrangements do not change the value of the sum. This is not true for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Series Acceleration
In mathematics, series acceleration is one of a collection of sequence transformations for improving the rate of convergence of a series. Techniques for series acceleration are often applied in numerical analysis, where they are used to improve the speed of numerical integration. Series acceleration techniques may also be used, for example, to obtain a variety of identities on special functions. Thus, the Euler transform applied to the hypergeometric series gives some of the classic, well-known hypergeometric series identities. Definition Given a sequence :S=\_ having a limit :\lim_ s_n = \ell, an accelerated series is a second sequence :S'=\_ which converges faster to \ell than the original sequence, in the sense that :\lim_ \frac = 0. If the original sequence is divergent, the sequence transformation acts as an extrapolation method to the antilimit \ell. The mappings from the original to the transformed series may be linear (as defined in the article sequence tr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Johnsonbaugh
Richard F. Johnsonbaugh (born 1941) is an American mathematician and computer scientist. His interests include discrete mathematics and the history of mathematics. He is the author of several textbooks. Johnsonbaugh earned a bachelor's degree in mathematics from Yale University, and then moved to the University of Oregon for graduate study.Author biography from ''Discrete Mathematics'' (8th ed.) He completed his Ph.D. at Oregon in 1969. His dissertation, ''I. Classical Fundamental Groups and Covering Space Theory in the Setting of Cartan and Chevalley; II. Spaces and Algebras of Vector-Valued Differentiable Functions'', was supervised by Bertram Yood. He also has a second master's degree in computer science from the University of Illinois at Chicago. He is currently professor emeritus at De Paul University DePaul University is a private university, private, Catholic higher education, Catholic research university in Chicago, Illinois. Founded by the Congregation of the Mission ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
