List Of Real Analysis Topics

	List Of Real Analysis Topics This is a list of articles that are considered real analysis topics. General topics Limits Limit of a sequence Subsequential limit – the limit of some subsequence Limit of a function (''see List of limits for a list of limits of common functions) One-sided limit – either of the two limits of functions of real variables x, as x approaches a point from above or below Squeeze theorem – confirms the limit of a function via comparison with two other functions ** Big O notation – used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions Sequences and series (''see also list of mathematical series'') Arithmetic progression – a sequence of numbers such that the difference between the consecutive terms is constant Generalized arithmetic progression – a sequence of numbers such that the difference between consecutive terms can be one of several possible constants Geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Real Analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability. Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions. Scope Construction of the real numbers The theorems of real analysis rely on the properties of the real number system, which must be established. The real number system consists of an uncountable set (\mathbb), together with two binary operations denoted and , and an order denoted . The operations make the real numbers a field, and, along with the order, an ordered field. The real number system is the unique '' complete ordered field'', in the sense that any other complete ordered field is isomorphic to it. Intuitively, completen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Limit Of A Sequence As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1." In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the \lim symbol (e.g., \lim_a_n).Courant (1961), p. 29. If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests. Limits can be defined in any metric or topological space, but are usually first encountered in the real numbers. History The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involve limiting processes. Leucippus, Democritus, Antiphon, Eudoxus, and Archimedes developed the method of exhaustion, which uses an infinite sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Harmonic Series (mathematics) In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: \sum_^\infty\frac = 1 + \frac + \frac + \frac + \frac + \cdots. The first n terms of the series sum to approximately \ln n + \gamma, where \ln is the natural logarithm and \gamma\approx0.577 is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it is a divergent series. Its divergence was proven in the 14th century by Nicole Oresme using a precursor to the Cauchy condensation test for the convergence of infinite series. It can also be proven to diverge by comparing the sum to an integral, according to the integral test for convergence. Applications of the harmonic series and its partial sums include Divergence of the sum of the reciprocals of the primes, Euler's proof that there are infinitely many prime numbers, the analysis of the coupon collector's problem on how many random trials are nee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Divergent Geometric Series In mathematics, an infinite geometric series of the form :\sum_^\infty ar^ = a + ar + ar^2 + ar^3 +\cdots is divergent if and only if , ''r'' , ≥ 1. Methods for summation of divergent series are sometimes useful, and usually evaluate divergent geometric series to a sum that agrees with the formula for the convergent case :\sum_^\infty ar^ = \frac. This is true of any summation method that possesses the properties of regularity, linearity, and stability. Examples In increasing order of difficulty to sum: * 1 − 1 + 1 − 1 + · · ·, whose common ratio is −1 * 1 − 2 + 4 − 8 + · · ·, whose common ratio is −2 1 + 2 + 4 + 8 + · · ·, whose common ratio is 2 1 + 1 + 1 + 1 + · · ·, whose common ratio is 1. Motivation for study It is useful to figure out which summation methods produce the geometric series formula for which common ratios. One application for this information is the so-called Borel-Okada principle: If a regular summation me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	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 ... [...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 hyper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Telescoping Series In mathematics, a telescoping series is a series whose general term t_n can be written as t_n=a_n-a_, i.e. the difference of two consecutive terms of a sequence (a_n). As a consequence the partial sums only consists of two terms of (a_n) after cancellation. The cancellation technique, with part of each term cancelling with part of the next term, is known as the method of differences. For example, the series :\sum_^\infty\frac (the series of reciprocals of pronic numbers) simplifies as :\begin \sum_^\infty \frac & = \sum_^\infty \left( \frac - \frac \right) \\ & = \lim_ \sum_^N \left( \frac - \frac \right) \\ & = \lim_ \left\lbrack \right\rbrack \\ & = \lim_ \left\lbrack \right\rbrack \\ & = \lim_ \left\lbrack \right\rbrack = 1. \end An early statement of the formula for the sum or partial sums of a telescoping series can be found in a 1644 work by Evangelista Torricelli, ''De dimensione parabolae''. In general Telescoping sums are finite sums in which pair ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Binomial Series In mathematics, the binomial series is a generalization of the polynomial that comes from a binomial formula expression like (1+x)^n for a nonnegative integer n. Specifically, the binomial series is the Taylor series for the function f(x)=(1+x)^ centered at x = 0, where \alpha \in \Complex and , x, 0 and diverges to +\infty if \operatorname\alpha<0. If $\operatorname\alpha=0$ , then $n^ = e^$ converges if and only if the sequence $\operatorname\alpha\log n$ converges $\bmod$ , which is certainly true if $\alpha=0$ but false if $\operatorname\alpha \neq0$ : in the latter case the sequence is dense $\bmod$ , due to the fact that $\log n$ diverges and $\log (n+1)-\log n$ converges to zero). Summation of the binomial series The usual argument to compute the sum of the binomial series goes as follows. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Power Series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a constant. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, ''c'' (the ''center'' of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form \sum_^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots. Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions (a kind of formal power series) and in electronic engineering (under the name of the Z-transform). The familiar decimal notation for real numbers can also be viewe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Divergent Series In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. A counterexample is the harmonic series :1 + \frac + \frac + \frac + \frac + \cdots =\sum_^\infty\frac. The divergence of the harmonic series was proven by the medieval mathematician Nicole Oresme. In specialized mathematical contexts, values can be objectively assigned to certain series whose sequences of partial sums diverge, in order to make meaning of the divergence of the series. A ''summability method'' or ''summation method'' is a partial function from the set of series to values. For example, Cesàro summation assigns Grandi's diverge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Convergent Series In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (a_0, a_1, a_2, \ldots) defines a series that is denoted :S=a_0 +a_1+ a_2 + \cdots=\sum_^\infty a_k. The th partial sum is the sum of the first terms of the sequence; that is, :S_n = \sum_^n a_k. A series is convergent (or converges) if the sequence (S_1, S_2, S_3, \dots) of its partial sums tends to a limit; that means that, when adding one a_k after the other ''in the order given by the indices'', one gets partial sums that become closer and closer to a given number. More precisely, a series converges, if there exists a number \ell such that for every arbitrarily small positive number \varepsilon, there is a (sufficiently large) integer N such that for all n \ge N, :\left , S_n - \ell \right , 1 produce a convergent series: : ++++++\cdots = . Alternating the signs of reciprocals of powers of 2 also produces a convergent series: : -+-+-+\cdo ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]

Summation of the binomial series