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Abel's Theorem
In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel, who proved it in 1826. Theorem Let the Taylor series G (x) = \sum_^\infty a_k x^k be a power series with real coefficients a_k with radius of convergence 1. Suppose that the series \sum_^\infty a_k converges. Then G(x) is continuous from the left at x = 1, that is, \lim_ G(x) = \sum_^\infty a_k. The same theorem holds for complex power series G(z) = \sum_^\infty a_k z^k, provided that z \to 1 entirely within a single ''Stolz sector'', that is, a region of the open unit disk where , 1-z, \leq M(1-, z, ) for some fixed finite M > 1. Without this restriction, the limit may fail to exist: for example, the power series \sum_ \frac n converges to 0 at z = 1, but is unbounded near any point of the form e^, so the value at z = 1 is not the limit as z tends to 1 in the whole open disk. Note that G(z ... [...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 ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a constant called the ''center'' of the series. 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, the center ''c'' is equal to zero, for instance for 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. The partial sums of a power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Convergence
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E as the function domain if, given any arbitrarily small positive number \varepsilon, a number N can be found such that each of the functions f_N, f_,f_,\ldots differs from f by no more than \varepsilon ''at every point'' x ''in'' E. Described in an informal way, if f_n converges to f uniformly, then how quickly the functions f_n approach f is "uniform" throughout E in the following sense: in order to guarantee that f_n(x) differs from f(x) by less than a chosen distance \varepsilon, we only need to make sure that n is larger than or equal to a certain N, which we can find without knowing the value of x\in E in advance. In other words, there exists a number N=N(\varepsilon) that could depend on \varepsilon but is ''independent of x'', such that choosing n\geq N wi ... [...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 divergent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tauberian Theorems
In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The original examples are Abel's theorem showing that if a series converges to some limit then its Abel sum is the same limit, and Tauber's theorem showing that if the Abel sum of a series exists and the coefficients are sufficiently small (o(1/''n'')) then the series converges to the Abel sum. More general Abelian and Tauberian theorems give similar results for more general summation methods. There is not yet a clear distinction between Abelian and Tauberian theorems, and no generally accepted definition of what these terms mean. Often, a theorem is called "Abelian" if it shows that some summation method gives the usual sum for convergent series, and is called "Tauberian" if it gives conditions for a series summable by some method that allows it to be summable in the usual sense. In th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Converse (logic)
In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the Material conditional, implication ''P'' → ''Q'', the converse is ''Q'' → ''P''. For the categorical proposition ''All S are P'', the converse is ''All P are S''. Either way, the truth of the converse is generally independent from that of the original statement.Robert Audi, ed. (1999), ''The Cambridge Dictionary of Philosophy'', 2nd ed., Cambridge University Press: "converse". Implicational converse Let ''S'' be a statement of the form ''P implies Q'' (''P'' → ''Q''). Then the ''converse'' of ''S'' is the statement ''Q implies P'' (''Q'' → ''P''). In general, the truth of ''S'' says nothing about the truth of its converse, unless the Antecedent (logic), antecedent ''P'' and the consequent ''Q'' are logically equivalent. For example, consider the true statement "If I am a human, then I am mortal." The converse of that stateme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Summation By Parts
In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation, named after Niels Henrik Abel who introduced it in 1826. Statement Suppose \ and \ are two sequences. Then, :\sum_^n f_k(g_-g_k) = \left(f_g_ - f_m g_m\right) - \sum_^n g_(f_- f_). Using the forward difference operator \Delta, it can be stated more succinctly as :\sum_^n f_k\Delta g_k = \left(f_ g_ - f_m g_m\right) - \sum_^ g_\Delta f_k, Summation by parts is an analogue to integration by parts: :\int f\,dg = f g - \int g\,df, or to Abel's summation formula: :\sum_^n f(k)(g_-g_)= \left(f(n)g_ - f(m) g_m\right) - \int_^n g_ f'(t) dt. An alternative statement is :f_n g_n - f_m g_m = \sum_^ f_k\Delta g_k + \sum_^ g_k\Delta f_k + \sum_^ \Delta f_k \Delta g_k which is analogous to the integration by parts formula for semimartingales ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galton–Watson Process
The Galton–Watson process, also called the Bienaymé-Galton-Watson process or the Galton-Watson branching process, is a branching stochastic process arising from Francis Galton's statistical investigation of the extinction of family names. The process models family names as patrilineal (passed from father to son), while offspring are randomly either male or female, and names become extinct if the family name line dies out (holders of the family name die without male descendants). Galton's investigation of this process laid the groundwork for the study of branching processes as a subfield of probability theory, and along with these subsequent processes the Galton-Watson process has found numerous applications across population genetics, computer science, and other fields. History There was concern amongst the Victorians that aristocratic surnames were becoming extinct. In 1869, Galton published '' Hereditary Genius'', in which he treated the extinction of different social ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability-generating Function
In probability theory, the probability generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable. Probability generating functions are often employed for their succinct description of the sequence of probabilities Pr(''X'' = ''i'') in the probability mass function for a random variable ''X'', and to make available the well-developed theory of power series with non-negative coefficients. Definition Univariate case If ''X'' is a discrete random variable taking values ''x'' in the non-negative integers , then the ''probability generating function'' of ''X'' is defined as G(z) = \operatorname (z^X) = \sum_^ p(x) z^x, where p is the probability mass function of X. Note that the subscripted notations G_X and p_X are often used to emphasize that these pertain to a particular random variable X, and to its distribution. The power series converges absolutely at least for all com ... [...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] |
Generating Function
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease with which they can be handled may differ considerably. The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed. Generating functions are sometimes called generating series, in that a series of terms can be said to be the generator of its sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Series
In mathematics, the binomial series is a generalization of the binomial formula to cases where the exponent is not a positive integer: where \alpha is any complex number, and the power series on the right-hand side is expressed in terms of the (generalized) binomial coefficients :\binom = \frac. The binomial series is the MacLaurin series for the function f(x)=(1+x)^\alpha. It converges when , x, - 1 is assumed. On the other hand, the series does not converge if , x, =1 and \operatorname(\alpha) \le - 1 , again by formula (). Alternatively, we may observe that for all j, \left, \fracj - 1 \ \ge 1 - \fracj \ge 1 . Thus, by formula (), for all k, \left, \ \ge 1 . This completes the proof of (iii). Turning to (iv), we use identity () above with x=-1 and \alpha-1 in place of \alpha, along with formula (), to obtain :\sum_^n \! (-1)^k = \! (-1)^n= \frac1 (1+o(1)) as n\to\infty. Assertion (iv) now follows from the asymptotic behavior of the sequence n^ = e^. (Precisely, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
One-sided Limit
In calculus, a one-sided limit refers to either one of the two Limit of a function, limits of a Function (mathematics), function f(x) of a Real number, real variable x as x approaches a specified point either from the left or from the right. The limit as x decreases in value approaching a (x approaches a "from the right" or "from above") can be denoted: \lim_f(x) \quad \text \quad \lim_\,f(x) \quad \text \quad \lim_\,f(x) \quad \text \quad f(x+) The limit as x increases in value approaching a (x approaches a "from the left" or "from below") can be denoted: \lim_f(x) \quad \text \quad \lim_\, f(x) \quad \text \quad \lim_\,f(x) \quad \text \quad f(x-) If the limit of f(x) as x approaches a exists then the limits from the left and from the right both exist and are equal. In some cases in which the limit \lim_ f(x) does not exist, the two one-sided limits nonetheless exist. Consequently, the limit as x approaches a is sometimes called a "two-sided limit". It is possible for exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
