In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity.[1] 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 of the race, the tortoise has reached a second position; when he reaches this second position, the tortoise is at a third position, and so on. Zeno concluded that Achilles could never reach the tortoise, and thus that movement does not exist. Zeno divided the race into infinitely many sub-races, each requiring a finite amount of time, so that the total time for Achilles to catch the tortoise is given by a series. The resolution of the paradox is that, although the series has an infinite number of terms, it has a finite sum, which gives the time necessary for Achilles to catch up with the tortoise.
02 Tv Series Apk Download
Download File 🔥 https://tlniurl.com/2yGc7M 🔥
The infinite sequence of additions implied by a series cannot be effectively carried on (at least in a finite amount of time). However, if the set to which the terms and their finite sums belong has a notion of limit, it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as n tends to infinity (if the limit exists) of the finite sums of the n first terms of the series, which are called the nth partial sums of the series. That is,
Generally, the terms of a series come from a ring, often the field R {\displaystyle \mathbb {R} } of the real numbers or the field C {\displaystyle \mathbb {C} } of the complex numbers. In this case, the set of all series is itself a ring (and even an associative algebra), in which the addition consists of adding the series term by term, and the multiplication is the Cauchy product.
An easy way that an infinite series can converge is if all the an 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.
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, 1/2, 1/4, 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 1/2, we still have a piece of length 1/2 unmarked, so we can certainly mark the next 1/4. 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. Given that the series converges, proving that it is equal to 2 requires only elementary algebra. If the series is denoted S, it can be seen that
Since 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, the decimal expansion 0.111... can be identified with 1/9. This leads to an argument that 9 0.111... = 0.999... = 1, which only relies on the fact that the limit laws for series preserve the arithmetic operations; for more detail on this argument, see 0.999....
Series are classified not only by whether they converge or diverge, but also by the properties of the terms an (absolute or conditional convergence); type of convergence of the series (pointwise, uniform); the class of the term an (whether it is a real number, arithmetic progression, trigonometric function); etc.
When you group a series reordering of the series does not happen, so Riemann series theorem does not apply. A new series will have its partial sums as subsequence of original series, which means if the original series converges, so does the new series. But for divergent series that is not true, for example 1-1+1-1+... grouped every two elements will create 0+0+0+... series, which is convergent. On the other hand, divergence of the new series means the original series can be only divergent which is sometimes useful, like in Oresme proof.
A series of real or complex numbers is said to be conditionally convergent (or semi-convergent) if it is convergent but not absolutely convergent. A famous example is the alternating series
A stronger notion of convergence of a series of functions is the uniform convergence. A series converges uniformly if it converges pointwise to the function tag_hash_120(x), and the error in approximating the limit by the Nth partial sum,
Uniform convergence is desirable for a series because many properties of the terms of the series are then retained by the limit. For example, if a series of continuous functions converges uniformly, then the limit function is also continuous. Similarly, if the tag_hash_123n are integrable on a closed and bounded interval I and converge uniformly, then the series is also integrable on I and can be integrated term-by-term. Tests for uniform convergence include the Weierstrass' M-test, Abel's uniform convergence test, Dini's test, and the Cauchy criterion.
More sophisticated types of convergence of a series of functions can also be defined. In measure theory, for instance, a series of functions converges almost everywhere if it converges pointwise except on a certain set of measure zero. Other modes of convergence depend on a different metric space structure on the space of functions under consideration. For instance, a series of functions converges in mean on a set E to a limit function tag_hash_128 provided
Unless it converges only at x=c, such a series converges on a certain open disc of convergence centered at the point c in the complex plane, and may also converge at some of the points of the boundary of the disc. The radius of this disc is known as the radius of convergence, and can in principle be determined from the asymptotics of the coefficients an. The convergence is uniform on closed and bounded (that is, compact) subsets of the interior of the disc of convergence: to wit, it is uniformly convergent on compact sets.
Historically, mathematicians such as Leonhard Euler operated liberally with infinite series, even if they were not convergent. When calculus was put on a sound and correct foundation in the nineteenth century, rigorous proofs of the convergence of series were always required.
Even if the limit of the power series is not considered, if the terms support appropriate structure then it is possible to define operations such as addition, multiplication, derivative, antiderivative for power series "formally", treating the symbol "+" as if it corresponded to addition. In the most common setting, the terms come from a commutative ring, so that the formal power series can be added term-by-term and multiplied via the Cauchy product. In this case the algebra of formal power series is the total algebra of the monoid of natural numbers over the underlying term ring.[12] If the underlying term ring is a differential algebra, then the algebra of formal power series is also a differential algebra, with differentiation performed term-by-term.
If such a series converges, then in general it does so in an annulus rather than a disc, and possibly some boundary points. The series converges uniformly on compact subsets of the interior of the annulus of convergence.
In the 17th century, James Gregory worked in the new decimal system 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.
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.
corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values of m {\displaystyle m} and x {\displaystyle x} . He showed the necessity of considering the subject of continuity in questions of convergence.
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). Schlmilch (Zeitschrift, Vol.I, p. 192, 1856) also improved Jacobi's remainder, and showed the relation between the remainder and Bernoulli's function
Fourier series were being investigatedas the result of physical considerations at the same time thatGauss, Abel, and Cauchy were working out the theory of infiniteseries. Series for the expansion of sines and cosines, of multiplearcs in powers of the sine and cosine of the arc had been treated byJacob Bernoulli (1702) and his brother Johann Bernoulli (1701) and stillearlier by Vieta. Euler and Lagrange simplified the subject,as did Poinsot, Schrter, Glaisher, and Kummer. 152ee80cbc
free fire rose emote video download
how to re download songs on amazon music