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:1 + 2 + 3 + 4 + 5 + ... + 99 + 100. In most cases of interest the terms of the sequence are produced according to a certain rule, such as by a Formula , by an Algorithm , by a sequence of Measurement s, or even by a Random Number Generator . A series may be Finite or ''infinite''. Finite series may be handled with elementary Algebra , but infinite series require tools from Mathematical Analysis if they are to be applied in anything more than a tentative way. Examples of simple series include the Arithmetic Series which is a sum of an Arithmetic Progression , written as: : and finite Geometric Series , a sum of a Geometric Progression , which can be written as: : INFINITE SERIES The sum of an infinite series is the Limit of the Sequence of '''partial sums''' : as ''n'' → ∞, if that limit exists. If the limit exists and is finite, the series is said to converge ; if it is infinite or does not exist, the series is said to '''diverge''' . The very simplest way that an infinite series can converge is if all the ''a''''n'' are zero for ''n'' sufficiently large. Such a series can be identified with a finite sum, so it is only infinite in a trivial sense. However, infinite series of nonzero terms can also converge, which resolves the mathematical side of several of Zeno's Paradoxes . The simplest case of a nontrivial infinite series is perhaps : It is possible to "visualize" its convergence on the Real Number Line : we can imagine a line of length 2, with successive segments marked off of lengths 1, ½, ¼, etc. There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: when we have marked off ½, we still have a piece of length ½ unmarked, so we can certainly mark the next ¼. This argument does not prove that the sum is ''equal'' to 2 (although it is), but it does prove that it is ''at most'' 2. In other words, the series has an upper bound. This series is a geometric series and mathematicians usually write it as: : An infinite series is formally written as : where the elements ''a''''n'' are real (or Complex ) numbers. We say that this series converges to ''S'', or that its sum is ''S'', if the Limit : exists and is equal to ''S''. If there is no such number, then the series is said to ''diverge''. Formal definition Mathematicians usually study a series as a ''pair'' of sequences: the sequence of terms of the series: and the sequence of '''partial sums''' where . The notation : represents then '' A Priori '' this pair of sequences, which is always well defined, but which may or may not converge. In the case of convergence, i.e., if the sequence of partial sums SN has a limit, the notation is also used to denote the ''limit'' of this sequence. To make a distinction between these two completely different objects (sequence vs. numerical value), one may sometimes omit the limits (atop and below the sum's symbol) in the former case, although it is usually clear from the context which one is meant. Also, different notions of convergence of such a sequence do exist ( Absolute Convergence , summability, etc). In case the elements of the sequence (and thus of the series) are not simple numbers, but, for example, Function s, still more types of convergence can be considered (pointwise convergence, uniform convergence, etc.; see below). Mathematicians extend this idiom to other, equivalent notions of series. For instance, when we talk about a Recurring Decimal , we are talking, in fact, just about the series for which it stands (0.1 + 0.01 + 0.001 + …). But because these series always converge to Real Numbers (because of what is called the Completeness Property of the real numbers), to talk about the series in this way is the same as to talk about the numbers for which they stand. In particular, it should offend no sensibilities if we make no distinction between 0.111… and 1/9. Less clear is the argument that , but it is not untenable when we consider that we can formalize the proof knowing only that limit laws preserve the arithmetic operations. See 0.999... for more. HISTORY OF THE THEORY OF INFINITE SERIES Development of infinite series The idea of an Infinite series expansion of a function was first conceived in India by Madhava in the 14th Century , who also developed the concepts of the Power Series , the Taylor Series , the Maclaurin Series , rational approximations of infinite series, and infinite Continued Fraction s. He discovered a number of infinite series, including the Taylor Series of the Trigonometric Function s of Sine , Cosine , Tangent and Arctangent , the Taylor series approximations of the sine and cosine functions, and the Power Series of the Radius , Diameter , Circumference , angle θ , π and π/4. His students and followers in the Kerala School further expanded his works with various other series expansions and approximations, until the 16th Century . In the 17th Century , James Gregory also worked on infinite series and published several Maclaurin Series . In 1715 , a general method for constructing the Taylor Series for all functions for which they exist was provided by Brook Taylor . Leonhard Euler in the 18th Century , developed the theory of Hypergeometric Series and Q-series . Convergence criteria The study of the Convergence criteria of a series began with Madhava in the 14th century, who developed Tests Of Convergence of infinite series, which his followers further developed at the Kerala School. In Europe however, the investigation of the validity of infinite series is considered to begin with Gauss in the 19th Century . Euler had already considered the Hypergeometric Series : on which Gauss published a memoir in 1812. It established simpler criteria of convergence, and the questions of remainders and the range of convergence. Cauchy (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with him begins the discovery of effective criteria. The terms ''convergence'' and ''divergence'' had been introduced long before by Gregory (1668). Leonhard Euler and Gauss had given various criteria, and Colin Maclaurin had anticipated some of Cauchy's discoveries. Cauchy advanced the theory of Power Series by his expansion of a complex Function in such a form. Abel (1826) in his memoir on the Binomial Series : corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values of and . He showed the necessity of considering the subject of continuity in questions of convergence. Cauchy's methods led to special rather than general criteria, and the same may be said of Raabe (1832), who made the first elaborate investigation of the subject, of De Morgan (from 1842), whose logarithmic test DuBois-Reymond (1873) and Pringsheim (1889) have shown to fail within a certain region; of Bertrand (1842), Bonnet (1843), Malmsten (1846, 1847, the latter without integration); Stokes (1847), Paucker (1852), Tchebichef (1852), and Arndt (1853). General criteria began with Kummer (1835), and have been studied by Eisenstein (1847), Weierstrass in his various contributions to the theory of functions, Dini (1867), DuBois-Reymond (1873), and many others. Pringsheim's (from 1889) memoirs present the most complete general theory. Uniform convergence The theory of Uniform Convergence was treated by Cauchy (1821), his limitations being pointed out by Abel, but the first to attack it successfully were Seidel and Stokes (1847-48). Cauchy took up the problem again (1853), acknowledging Abel's criticism, and reaching the same conclusions which Stokes had already found. Thomae used the doctrine (1866), but there was great delay in recognizing the importance of distinguishing between uniform and non-uniform convergence, in spite of the demands of the theory of functions. Semi-convergence A series is said to be semi-convergent (or conditionally convergent) if it is convergent but not Absolutely Convergent . Semi-convergent series were studied by Poisson (1823), who also gave a general form for the remainder of the Maclaurin formula. The most important solution of the problem is due, however, to Jacobi (1834), who attacked the question of the remainder from a different standpoint and reached a different formula. This expression was also worked out, and another one given, by Malmsten (1847). Schlömilch (''Zeitschrift'', Vol.I, p. 192, 1856) also improved Jacobi's remainder, and showed the relation between the remainder and Bernoulli's Function : Genocchi (1852) has further contributed to the theory. Among the early writers was Wronski , whose "loi suprême" (1815) was hardly recognized until Cayley (1873) brought it into prominence. Fourier series Fourier Series were being investigated as the result of physical considerations at the same time that Gauss, Abel, and Cauchy were working out the theory of infinite series. Series for the expansion of sines and cosines, of multiple arcs in powers of the sine and cosine of the arc had been treated by Jakob Bernoulli (1702) and his brother Johann Bernoulli (1701) and still earlier by Viète . Euler and Lagrange simplified the subject, as did Poinsot , Schröter , Glaisher , and Kummer . Fourier (1807) set for himself a different problem, to expand a given function of ''x'' in terms of the sines or cosines of multiples of ''x'', a problem which he embodied in his '' Théorie Analytique De La Chaleur '' (1822). Euler had already given the formulas for determining the coefficients in the series; Fourier was the first to assert and attempt to prove the general theorem. Poisson (1820-23) also attacked the problem from a different standpoint. Fourier did not, however, settle the question of convergence of his series, a matter left for Cauchy (1826) to attempt and for Dirichlet (1829) to handle in a thoroughly scientific manner (see Convergence Of Fourier Series ). Dirichlet's treatment ('' Crelle '', 1829), of trigonometric series was the subject of criticism and improvement by Riemann (1854), Heine, Lipschitz , Schläfli , and DuBois-Reymond . Among other prominent contributors to the theory of trigonometric and Fourier series were Dini , Hermite , Halphen , Krause, Byerly and Appell . SOME TYPES OF INFINITE SERIES
:: :In general, the geometric series :: | ||
| :<math>\sum {n | 0}^\infty \lefta_n
ight</math> |
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| "http://wwwinformationdelightinfo/information/entry/Comparison_test" class="copylinks">Comparison Test 2: If ∑''b<sub>n</sub>''  is an absolutely convergent series such that ''a<sub>n+1</sub>'' /''a<sub>n</sub>''  ≤ ''C'' ''b<sub>n+1</sub>'' /''b<sub>n</sub>''  for some number ''C''  and for sufficiently large ''n'' , then ∑''a<sub>n</sub>''  converges absolutely as well If ∑''b<sub>n</sub>''  diverges, and ''a<sub>n+1</sub>'' /''a<sub>n</sub>''  ≥ ''b<sub>n+1</sub>'' /''b<sub>n</sub>''  for all sufficiently large ''n'' , then ∑''a<sub>n</sub>''  also fails to converge absolutely (though it could still be conditionally convergent, eg if the ''a<sub>n</sub>''  alternate in sign) |
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| "http://wwwinformationdelightinfo/information/entry/Ratio_test" class="copylinks">Ratio Test : If ''a''<sub>''n''+1</sub>/''a''<sub>''n''</sub> approaches a number less than one as n approaches infinity, then ∑ ''a''<sub>''n''</sub> converges absolutely When the ratio is 1, convergence can sometimes be determined as well |
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| "http://wwwinformationdelightinfo/information/entry/Root_test" class="copylinks">Root Test : If there exists a constant ''C'' < 1 such that ''a''<sub>''n''</sub><sup>1/''n''</sup> ≤ ''C'' for all sufficiently large ''n'', then ∑ ''a''<sub>''n''</sub> converges absolutely |
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| :<math>\sum {i\in I}a I | \lim_F\left\{\sum_{i\in A}a_i\,\biggA\in F
ight\}</math> |
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| :<math>\lim {n O\infty}\sum {i | 1}^n \a_i\</math> |
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| :<math>I N | \left\{i\in I \,\bigg a_i> rac{1}{n}
ight\}</math> |
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