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Root Test
In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series. It depends on the quantity :\limsup_\sqrt where a_n are the terms of the series, and states that the series converges absolutely if this quantity is less than one, but diverges if it is greater than one. It is particularly useful in connection with power series. Root test explanation The root test was developed first by Augustin-Louis Cauchy who published it in his textbook Cours d'analyse (1821). Thus, it is sometimes known as the Cauchy root test or Cauchy's radical test. For a series :\sum_^\infty a_n. the root test uses the number :C = \limsup_\sqrt where "lim sup" denotes the limit superior, possibly ∞+. Note that if :\lim_\sqrt converges then it equals ''C'' and may be used in the root test instead. The root test states that: * if ''C'' 1 then the series diverges, * if ''C'' = 1 and the limit approaches strictly from above then the series diverges, ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy–Hadamard Theorem
In mathematics, the Cauchy–Hadamard theorem is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a power series. It was published in 1821 by Cauchy, but remained relatively unknown until Hadamard rediscovered it. Hadamard's first publication of this result was in 1888; he also included it as part of his 1892 Ph.D. thesis. Theorem for one complex variable Consider the formal power series in one complex variable ''z'' of the form f(z) = \sum_^ c_ (z-a)^ where a, c_n \in \Complex. Then the radius of convergence R of ''f'' at the point ''a'' is given by \frac = \limsup_ \left( , c_ , ^ \right) where denotes the limit superior, the limit as approaches infinity of the supremum of the sequence values after the ''n''th position. If the sequence values are unbounded so that the is ∞, then the power series does not converge near , while if the is 0 then the radius of convergence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. : Suppose that there exists r such that :: \lim_\left, \frac\ = r. : If ''r'' 1, then the series diverges. If ''r'' = 1, the ratio test is inconclusive, and the series may converge or diverge. Root test This is also known as the ''n''th root test or Cauchy's criterion. : Let :: r=\limsup_\sqrt : where \limsup denotes the limit superior (possib ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ratio Test
In mathematics, the ratio test is a test (or "criterion") for the convergence of a series :\sum_^\infty a_n, where each term is a real or complex number and is nonzero when is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test. The test The usual form of the test makes use of the limit The ratio test states that: * if ''L'' 1 then the series diverges; * if ''L'' = 1 or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case. It is possible to make the ratio test applicable to certain cases where the limit ''L'' fails to exist, if limit superior and limit inferior are used. The test criteria can also be refined so that the test is sometimes conclusive even when ''L'' = 1. More specifically, let :R = \lim\sup \left, \frac\ :r = \lim\inf \left, \frac\. Then the ratio test states that: * if ''R'' 1, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Test For Convergence
In mathematics, the integral test for convergence is a method used to test infinite series of monotonous terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test. Statement of the test Consider an integer and a function defined on the unbounded interval , on which it is monotone decreasing. Then the infinite series :\sum_^\infty f(n) converges to a real number if and only if the improper integral :\int_N^\infty f(x)\,dx is finite. In particular, if the integral diverges, then the series diverges as well. Remark If the improper integral is finite, then the proof also gives the lower and upper bounds for the infinite series. Note that if the function f(x) is increasing, then the function -f(x) is decreasing and the above theorem applies. Proof The proof basically uses the comparison test, comparing the term with the integral of over the intervals and , respectively. The monotonous ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if the base is implicit, simply . Parentheses are sometimes added for clarity, giving , , or . This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity. The natural logarithm of is the power to which would have to be raised to equal . For example, is , because . The natural logarithm of itself, , is , because , while the natural logarithm of is , since . The natural logarithm can be defined for any positive real number as the area under the curve from to (with the area being negative when ). The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural". The definition of the natural logarithm can then b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iteration
Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. In mathematics and computer science, iteration (along with the related technique of recursion) is a standard element of algorithms. Mathematics In mathematics, iteration may refer to the process of iterating a function, i.e. applying a function repeatedly, using the output from one iteration as the input to the next. Iteration of apparently simple functions can produce complex behaviors and difficult problems – for examples, see the Collatz conjecture and juggler sequences. Another use of iteration in mathematics is in iterative methods which are used to produce approximate numerical solutions to certain mathematical problems. Newton's method is an example of an iterative method. Manual calculation of a number's square root is a co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ratio Test
In mathematics, the ratio test is a test (or "criterion") for the convergence of a series :\sum_^\infty a_n, where each term is a real or complex number and is nonzero when is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test. The test The usual form of the test makes use of the limit The ratio test states that: * if ''L'' 1 then the series diverges; * if ''L'' = 1 or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case. It is possible to make the ratio test applicable to certain cases where the limit ''L'' fails to exist, if limit superior and limit inferior are used. The test criteria can also be refined so that the test is sometimes conclusive even when ''L'' = 1. More specifically, let :R = \lim\sup \left, \frac\ :r = \lim\inf \left, \frac\. Then the ratio test states that: * if ''R'' 1, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ratio Test
In mathematics, the ratio test is a test (or "criterion") for the convergence of a series :\sum_^\infty a_n, where each term is a real or complex number and is nonzero when is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test. The test The usual form of the test makes use of the limit The ratio test states that: * if ''L'' 1 then the series diverges; * if ''L'' = 1 or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case. It is possible to make the ratio test applicable to certain cases where the limit ''L'' fails to exist, if limit superior and limit inferior are used. The test criteria can also be refined so that the test is sometimes conclusive even when ''L'' = 1. More specifically, let :R = \lim\sup \left, \frac\ :r = \lim\inf \left, \frac\. Then the ratio test states that: * if ''R'' 1, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ratio Test
In mathematics, the ratio test is a test (or "criterion") for the convergence of a series :\sum_^\infty a_n, where each term is a real or complex number and is nonzero when is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test. The test The usual form of the test makes use of the limit The ratio test states that: * if ''L'' 1 then the series diverges; * if ''L'' = 1 or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case. It is possible to make the ratio test applicable to certain cases where the limit ''L'' fails to exist, if limit superior and limit inferior are used. The test criteria can also be refined so that the test is sometimes conclusive even when ''L'' = 1. More specifically, let :R = \lim\sup \left, \frac\ :r = \lim\inf \left, \frac\. Then the ratio test states that: * if ''R'' 1, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each successive term can be obtained by multiplying the previous term by 1/2. In general, a geometric series is written as a + ar + ar^2 + ar^3 + ..., where a is the coefficient of each term and r is the common ratio between adjacent terms. The geometric series had an important role in the early development of calculus, is used throughout mathematics, and can serve as an introduction to frequently used mathematical tools such as the Taylor series, the complex Fourier series, and the matrix exponential. The name geometric series indicates each term is the geometric mean of its two neighboring terms, similar to how the name arithmetic series indicates each term is the arithmetic mean of its two neighboring terms. The sequence of geometric series term ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
