# data[].attributes.headline
طرق التسويق الناجح 

# data[].attributes.subject
طرق التسويق الناجح 

# data[].attributes.attachment_link.title
افضل اساليب الترويج والتسويق للشركات الصغيرة

# data[].attributes.headline
فوائد السطوح المنحدرة

# data[].attributes.subject
فوائد السطوح المنحدرة

# data[].attributes.body
http://www.phys4arab.net/vb/showthread.php?t=59566

# data[].attributes.headline
حقائق عن الطقس

# data[].attributes.subject
حقائق عن الطقس

# data[].attributes.body
http://www.lovely0smile.com/Cat-49-0.html

# data[].attributes.headline
الكسور في حياتنا

# data[].attributes.subject
الكسور في حياتنا

# data[].attributes.attachment_link.authors[].name
الرياضيات في حياتنا

# data[].attributes.attachment_link.title
الكسور في حياتنا

# data[].attributes.headline
الهيئة العامة للإحصاء في المملكه

# data[].attributes.subject
الهيئة العامة للإحصاء في المملكه

# data[].attributes.body
https://www.stats.gov.sa/

# data[].attributes.headline
متتابعة فيبوناتشي

# data[].attributes.subject
متتابعة فيبوناتشي

# data[].attributes.attachment_link.title
متتابعة فيبوناتشي.

# data[].attributes.headline
المتتابعات المشهورة بالانجليزي

# data[].attributes.subject
المتتابعات المشهورة بالانجليزي

# data[].attributes.attachment_link.readable_text
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.

In modern terminology, any (ordered) infinite sequence of terms (that is, numbers, functions, or anything that can be added) defines a series, which is the operation of adding the a i one after the other. To emphasize that there are an infinite number of terms, a series may be called an infinite series. Such a series is represented (or denoted) by an expression like

or, using the summation sign,

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,

When this limit exists, one says that the series is convergent or summable, or that the sequence is summable. In this case, the limit is called the sum of the series. Otherwise, the series is said to be divergent.[2]

The notation denotes both the series-that is the implicit process of adding the terms one after the other indefinitely-and, if the series is convergent, the sum of the series-the result of the process. This is a generalization of the similar convention of denoting by both the addition-the process of adding-and its result-the sum of a and b.

Generally, the terms of a series come from a ring, often the field of the real numbers or the field 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 infinite series or simply a series is an infinite sum, represented by an infinite expression of the form[3]

where is any ordered sequence of terms, such as numbers, functions, or anything else that can be added (an abelian group). This is an expression that is obtained from the list of terms by laying them side by side, and conjoining them with the symbol "+". A series may also be represented by using summation notation, such as

If an abelian group A of terms has a concept of limit (e.g., if it is a metric space), then some series, the convergent series, can be interpreted as having a value in A, called the sum of the series. This includes the common cases from calculus, in which the group is the field of real numbers or the field of complex numbers. Given a series , its kth partial sum is[2]

By definition, the series converges to the limit L (or simply sums to L), if the sequence of its partial sums has a limit L.[3] In this case, one usually writes

A series is said to be convergent if it converges to some limit, or divergent when it does not. The value of this limit, if it exists, is then the value of the series.

Illustration of 3 geometric series with partial sums from 1 to 6 terms. The dashed line represents the limit.

A series Σ( a n is said to converge or to s k) of partial sums has a finite s k is infinite or does not exist, the series is said to be convergent when the sequence limit. If the limit of diverge.[4][2] When the limit of partial sums exists, it is called the value (or sum) of the series

An easy way that an infinite series can converge is if all the a n sufficiently large. Such a series can be identified with a finite sum, so it is only infinite in a trivial sense.n are zero for

Working out the properties of the series that converge, even if infinitely many terms are nonzero, is the essence of the study of series. Consider the example

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

Therefore,

The idiom can be extended to other, equivalent notions of series. For instance, a recurring decimal, as in

encodes the series

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....

A geometric series is one where each successive term is produced by multiplying the previous term by a constant number (called the common ratio in this context). For example:[2]

In general, the geometric series

converges if and only if , in which case it converges to .

The harmonic series is the series[5]

The harmonic series is divergent.

An alternating series is a series where terms alternate signs. Examples:

( alternating harmonic series) and

A telescoping series

converges if the sequence b n converges to a limit L-as n goes to infinity. The value of the series is then b ₁ − L.

An arithmetico-geometric series is a generalization of the geometric series, which has coefficients of the common ratio equal to the terms in an arithmetic sequence. Example:

The p-series

converges for p > 1 and diverges for p ≤ 1, which can be shown with the integral criterion described below in convergence tests. As a function of p, the sum of this series is Riemann's zeta function.

Hypergeometric series:

and their generalizations (such as basic hypergeometric series and elliptic hypergeometric series) frequently appear in integrable systems and mathematical physics.[6]

There are some elementary series whose convergence is not yet known/proven. For example, it is unknown whether the Flint Hills series

converges or not. The convergence depends on how well can be approximated with rational numbers (which is unknown as of yet). More specifically, the values of n with large numerical contributions to the sum are the numerators of the continued fraction convergents of , a sequence beginning with 1, 3, 22, 333, 355, 103993, ... (sequence A046947 in the OEIS). These are integers n that are close to for some integer m, so that is close to and its reciprocal is large.

[2]

Natural logarithm base e

Partial summation takes as input a sequence, ( a n), and gives as output another sequence, ( S N). It is thus a unary operation on sequences. Further, this function is linear, and thus is a linear operator on the vector space of sequences, denoted Σ. The inverse operator is the finite difference operator, denoted Δ. These behave as discrete analogues of integration and differentiation, only for series (functions of a natural number) instead of functions of a real variable. For example, the sequence (1, 1, 1, ...) has series (1, 2, 3, 4, ...) as its partial summation, which is analogous to the fact that

In computer science, it is known as prefix sum.

Series are classified not only by whether they converge or diverge, but also by the properties of the terms a n (absolute or conditional convergence); type of convergence of the series (pointwise, uniform); the class of the term a n (whether it is a real number, arithmetic progression, trigonometric function); etc.

When an is a non-negative real number for every n, the sequence SN of partial sums is non-decreasing. It follows that a series Σ an with non-negative terms converges if and only if the sequence SN of partial sums is bounded.

For example, the series

is convergent, because the inequality

and a telescopic sum argument implies that the partial sums are bounded by 2. The exact value of the original series is the Basel problem.

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

converges absolutely if the series of absolute values

converges. This is sufficient to guarantee not only that the original series converges to a limit, but also that any reordering of it converges to the same limit.

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

which is convergent (and its sum is equal to ), but the series formed by taking the absolute value of each term is the divergent harmonic series. The Riemann series theorem says that any conditionally convergent series can be reordered to make a divergent series, and moreover, if the are real and is any real number, that one can find a reordering so that the reordered series converges with sum equal to .

Abel's test is an important tool for handling semi-convergent series. If a series has the form

where the partial sums are bounded, has bounded variation, and exists:

with . Abel's method consists in writing , and in performing a transformation similar to integration by parts (called summation by parts), that relates the given series to the absolutely convergent series

The evaluation of truncation errors is an important procedure in numerical analysis (especially validated numerics and computer-assisted proof).

When conditions of the alternating series test are satisfied by , there is an exact error evaluation.[7] Set to be the partial sum of the given alternating series . Then the next inequality holds:

Taylor's theorem is a statement that includes the evaluation of the error term when the Taylor series is truncated.

By using the ratio, we can obtain the evaluation of the error term when the hypergeometric series is truncated.[8]

For the matrix exponential:

the following error evaluation holds (scaling and squaring method):[9][10][11]

There exist many tests that can be used to determine whether particular series converge or diverge.

n-th term test: If , then the series diverges; if , then the test is inconclusive.

Comparison test 1 (see Direct comparison test): If is an absolutely convergent series such that for some number and for sufficiently large , then converges absolutely as well. If diverges, and for all sufficiently large , then also fails to converge absolutely (though it could still be conditionally convergent, for example, if the alternate in sign).

Comparison test 2 (see Limit comparison test): If is an absolutely convergent series such that for sufficiently large , then converges absolutely as well. If diverges, and for all sufficiently large , then also fails to converge absolutely (though it could still be conditionally convergent, for example, if the alternate in sign).

Ratio test: If there exists a constant such that for all sufficiently large , then converges absolutely. When the ratio is less than , but not less than a constant less than , convergence is possible but this test does not establish it.

Root test: If there exists a constant such that for all sufficiently large , then converges absolutely.

Integral test: if is a positive monotone decreasing function defined on the interval with for all , then converges if and only if the integral is finite.

Cauchy's condensation test: If is non-negative and non-increasing, then the two series and are of the same nature: both convergent, or both divergent.

Alternating series test: A series of the form (with ) is called alternating. Such a series converges if the sequence is monotone decreasing and converges to . The converse is in general not true.

For some specific types of series there are more specialized convergence tests, for instance for Fourier series there is the Dini test.

A series of real- or complex-valued functions

converges pointwise on a set E, if the series converges for each x in E as an ordinary series of real or complex numbers. Equivalently, the partial sums

converge to ƒ( x) as N → ∞ for each x ∈ E.

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 ƒ( x), and the error in approximating the limit by the N th partial sum,

can be made minimal independently of x by choosing a sufficiently large N.

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 ƒ n 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 ƒ provided

as N → ∞.

A power series is a series of the form

The Taylor series at a point c of a function is a power series that, in many cases, converges to the function in a neighborhood of c. For example, the series

is the Taylor series of at the origin and converges to it for every x.

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 a n. 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.

While many uses of power series refer to their sums, it is also possible to treat power series as formal sums, meaning that no addition operations are actually performed, and the symbol "+" is an abstract symbol of conjunction which is not necessarily interpreted as corresponding to addition. In this setting, the sequence of coefficients itself is of interest, rather than the convergence of the series. Formal power series are used in combinatorics to describe and study sequences that are otherwise difficult to handle, for example, using the method of generating functions. The Hilbert-Poincaré series is a formal power series used to study graded algebras.

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.

Laurent series generalize power series by admitting terms into the series with negative as well as positive exponents. A Laurent series is thus any series of the form

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.

A Dirichlet series is one of the form

where s is a complex number. For example, if all a n are equal to 1, then the Dirichlet series is the Riemann zeta function

Like the zeta function, Dirichlet series in general play an important role in analytic number theory. Generally a Dirichlet series converges if the real part of s is greater than a number called the abscissa of convergence. In many cases, a Dirichlet series can be extended to an analytic function outside the domain of convergence by analytic continuation. For example, the Dirichlet series for the zeta function converges absolutely when Re( s) > 1, but the zeta function can be extended to a holomorphic function defined on with a simple pole at 1.

This series can be directly generalized to general Dirichlet series.

A series of functions in which the terms are trigonometric functions is called a trigonometric series:

The most important example of a trigonometric series is the Fourier series of a function.

Greek mathematician Archimedes produced the first known summation of an infinite series with a method that is still used in the area of calculus today. He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of π.[13][14]

Mathematicians from the Kerala school were studying infinite series c. 1350 CE.[15]

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.

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), Chebyshev (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 memoirs (1889) present the most complete general theory.

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.

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 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 Jacob Bernoulli (1702) and his brother Johann Bernoulli (1701) and still earlier by Vieta. 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 du Bois-Reymond. Among other prominent contributors to the theory of trigonometric and Fourier series were Dini, Hermite, Halphen, Krause, Byerly and Appell.

Asymptotic series, otherwise asymptotic expansions, are infinite series whose partial sums become good approximations in the limit of some point of the domain. In general they do not converge, but they are useful as sequences of approximations, each of which provides a value close to the desired answer for a finite number of terms. The difference is that an asymptotic series cannot be made to produce an answer as exact as desired, the way that convergent series can. In fact, after a certain number of terms, a typical asymptotic series reaches its best approximation; if more terms are included, most such series will produce worse answers.

Under many circumstances, it is desirable to assign a limit to a series which fails to converge in the usual sense. A summability method is such an assignment of a limit to a subset of the set of divergent series which properly extends the classical notion of convergence. Summability methods include Cesàro summation, ( C, k) summation, Abel summation, and Borel summation, in increasing order of generality (and hence applicable to increasingly divergent series).

A variety of general results concerning possible summability methods are known. The Silverman-Toeplitz theorem characterizes matrix summability methods, which are methods for summing a divergent series by applying an infinite matrix to the vector of coefficients. The most general method for summing a divergent series is non-constructive, and concerns Banach limits.

Definitions may be given for sums over an arbitrary index set [16] There are two main differences with the usual notion of series: first, there is no specific order given on the set ; second, this set may be uncountable. The notion of convergence needs to be strengthened, because the concept of conditional convergence depends on the ordering of the index set.

If is a function from an index set to a set then the "series" associated to is the formal sum of the elements over the index elements denoted by the

When the index set is the natural numbers the function is a sequence denoted by A series indexed on the natural numbers is an ordered formal sum and so we rewrite as in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers

When the supremum is finite then the set of such that is countable. Indeed, for every the cardinality of the set is finite because

If is countably infinite and enumerated as then the above defined sum satisfies

provided the value is allowed for the sum of the series.
Any sum over non-negative reals can be understood as the integral of a non-negative function with respect to the counting measure, which accounts for the many similarities between the two constructions.

Let be a map, also denoted by from some non-empty set into a Hausdorff abelian topological group Let be the collection of all finite subsets of with viewed as a directed set, ordered under inclusion with union as join. The family is said to be unconditionally summable if the following limit, which is denoted by and is called the sum of exists in

Saying that the sum is the limit of finite partial sums means that for every neighborhood of the origin in there exists a finite subset of such that

Because is not totally ordered, this is not a limit of a sequence of partial sums, but rather of a net.[17][18]

For every neighborhood of the origin in there is a smaller neighborhood such that It follows that the finite partial sums of an unconditionally summable family form a Cauchy net, that is, for every neighborhood of the origin in there exists a finite subset of such that

which implies that for every (by taking and ).
When is complete, a family is unconditionally summable in if and only if the finite sums satisfy the latter Cauchy net condition. When is complete and is unconditionally summable in then for every subset the corresponding subfamily is also unconditionally summable in

If a family in is unconditionally summable then for every neighborhood of the origin in there is a finite subset such that for every index not in If is a first-countable space then it follows that the set of such that is countable. This need not be true in a general abelian topological group (see examples below).

Suppose that If a family is unconditionally summable in a Hausdorff abelian topological group then the series in the usual sense converges and has the same sum,

By nature, the definition of unconditional summability is insensitive to the order of the summation. When is unconditionally summable, then the series remains convergent after any permutation of the set of indices, with the same sum,

Conversely, if every permutation of a series converges, then the series is unconditionally convergent. When is complete then unconditional convergence is also equivalent to the fact that all subseries are convergent; if is a Banach space, this is equivalent to say that for every sequence of signs , the series

If is a topological vector space (TVS) and is a (possibly uncountable) family in then this family is summable if the limit of the net exists in where is the directed set of all finite subsets of directed by inclusion and

It is called absolutely summable if in addition, for every continuous seminorm on the family is summable. If is a normable space and if is an absolutely summable family in then necessarily all but a countable collection of 's are zero. Hence, in normed spaces, it is usually only ever necessary to consider series with countably many terms.

Summable families play an important role in the theory of nuclear spaces.

Series in Banach and seminormed spaces

The notion of series can be easily extended to the case of a seminormed space. If is a sequence of elements of a normed space and if then the series converges to in if the sequence of partial sums of the series converges to in ; to wit,

More generally, convergence of series can be defined in any abelian Hausdorff topological group. Specifically, in this case, converges to if the sequence of partial sums converges to

If is a seminormed space, then the notion of absolute convergence becomes: A series of vectors in converges absolutely if

If a countable series of vectors in a Banach space converges absolutely then it converges unconditionally, but the converse only holds in finite-dimensional Banach spaces (theorem of Dvoretzky & Rogers (1950)).

Conditionally convergent series can be considered if is a well-ordered set, for example, an ordinal number In this case, define by transfinite recursion:

1. Given a function into an abelian topological group define for every

a function whose support is a singleton Then

in the topology of pointwise convergence (that is, the sum is taken in the infinite product group ).

2. In the definition of partitions of unity, one constructs sums of functions over arbitrary index set

While, formally, this requires a notion of sums of uncountable series, by construction there are, for every given only finitely many nonzero terms in the sum, so issues regarding convergence of such sums do not arise. Actually, one usually assumes more: the family of functions is locally finite, that is, for every there is a neighborhood of in which all but a finite number of functions vanish. Any regularity property of the such as continuity, differentiability, that is preserved under finite sums will be preserved for the sum of any subcollection of this family of functions.

3. On the first uncountable ordinal viewed as a topological space in the order topology, the constant function given by satisfies

(in other words, copies of 1 is ) only if one takes a limit over all countable partial sums, rather than finite partial sums. This space is not separable.

Bromwich, T. J. An Introduction to the Theory of Infinite Series MacMillan & Co. 1908, revised 1926, reprinted 1939, 1942, 1949, 1955, 1959, 1965.

Dvoretzky, Aryeh; Rogers, C. Ambrose (1950). "Absolute and unconditional convergence in normed linear spaces". Proc. Natl. Acad. Sci. U.S.A. 36 (3): 192-197. Bibcode:1950PNAS...36..192D. doi:10.1073/pnas.36.3.192. PMC . PMID 16588972.

Swokowski, Earl W. (1983), Calculus with analytic geometry (Alternate ed.), Boston: Prindle, Weber & Schmidt, ISBN 978-0-87150-341-1

Walter Rudin, Principles of Mathematical Analysis (McGraw-Hill: New York, 1964).

Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.

Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.

# data[].attributes.attachment_link.description
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.

# data[].attributes.headline
مفارقات زينون الايلي

# data[].attributes.subject
مفارقات زينون الايلي

# data[].attributes.headline
فوائد الاقمار الصناعية

# data[].attributes.subject
فوائد الاقمار الصناعية

# data[].attributes.attachment_link.title
الأقمار الصناعية : إستخداماتها و أنواعها و المدارات التى تدور فيها

# data[].attributes.headline
كيف تحدد هدفك قبل فوات الاوان

# data[].attributes.subject
كيف تحدد هدفك قبل فوات الاوان

# data[].attributes.attachment_link.authors[].name
Samira - اقرألى كتاباً

# data[].attributes.attachment_link.title
الهدف هو أساس النجاح في الحياة 😍 إعرف هدفك قبل فوات الاوان 🤚

# data[].attributes.headline
ماهي وحدات القياس

# data[].attributes.subject
ماهي وحدات القياس

# data[].attributes.attachment_link.readable_text
القياس: هي مُقارنة بين شيئين أو أكثر، كمقارنة طول شخصين، حيث يُقاس طول كل شخص باستخدام الأداة المناسبة، ليُبنى بعدها الحُكم حول أيهما أطول من الآخر. أما وحدات القياس: فهي تُحوّل ما تم مقارنته إلى كميات معروفة، ومن أمثلة الكميات المُراد قياسها ما يأتي: النقد، والوقت، والطول والكتلة، والقوة، والحجم، والحرارة، وأيضاً الضغط، والطاقة.

قواعد يجب مراعاتها عند عملية القياس

هنالك عدة قواعد يجب أن تؤخذ بعين الاعتبار عند عملية القياس، ومنها القواعد الثلاثة الآتية:

يجب أن تكون الوحدة المستخدمة في القياس مناسبة لطبيعة الشيء الذي تم قياسه، فمثلاً وحدات الطول تُستخدم لقياس طول شيءٍ ما، أما وحدات الزمن فهي تُستخدم لقياس وقتٍ ما...وهكذا.

في بعض الأحيان تكون وحدات القياس غير كافية للحصول على قياساتٍ دقيقةٍ للغاية، لذلك نلجأ لاستخدام أجزائها، فمثلاً عند قياس طول جسم بالسنتمترات سيكون ذلك أكثر دقةً من قياس جسم مقرب بالديسيمترات، حيث يُعتبر القياس بمنزلة عشرية واحدة أكثر دقةً من القياس المقرّب لأقرب عدد صحيح.

أن تكون نتيجة القياس مناسبةً للأداة المستخدمة في القياس، فعند قياس طول جسم باستخدام المسطرة المجزّئة للسنتميترات، فلا بد من أن تكون نتيجة القياس بالستنمترات، حيث لا يجوز أن يكون ناتج القياس بوحدة غير موجودة بالأداة التي تم استخدامها في القياس.

إن للنقد دور كبير في حياتنا العملية، فهو الأداة التي يتعامل بها الأشخاص في المعاملات التجارية وعمليات البيع والشراء، ويُستخدم في الأردن الدينار الأردني كوحدة أَساسية لمعاملات البيع والشراء، ويمكن مضاعفة الدينار للحصول على فئات نقدية أُخرى، منها فئة الخمسة دنانير، وفئة العشرة دنانير، وكذلك فئة العشرين والخمسين ديناراً. أما بالنسبة لوحدات النقد العالمية المستخدمة فمنها الدولار، والفرنك، واليورو ، حيث إن لكل عملة من هذه العملات مضاعفات وأجزاء.

أما أجزاء الدينار فهي عبارة عن القطع النقدية التي تُمثل نِصفه ورُبعه، أيضاً قطع العشرة قروش، والخمسة قروش، والقرش، والفلس، ولتحويل إحداها إلى الأخرى لابد من معرفة العلاقة بين فئة الدينار وأجزائها، حيث يساعد ذلك على تسهيل أمور الحياة اليومية التي تخص التبادل ومعاملات البيع والشراء، ومنها ما يأتي:

الدينار= 100 قرشٍ.

نصف الدينار= 50 قرشاً.

ربع دينار= 25 قرشاً.

القرش= 10 فلسات.

يُعد الوقت من الموارد المهمة والرئيسية في تنظيم أمور الحياة، ويُقاس الزمن بوحداتٍ عدةٍ وهي: القرون، العقود، السنوات، الأيام، الساعات، الدقائق، الثواني.

القرن الواحد= 100 سنة.

العقد الواحد= 10 سنوات.

السنة الواحدة= 365 يوماً.

اليوم الواحد= 24 ساعة.

الساعة الواحدة= 60 دقيقة.

الدقيقة الواحدة= 60 ثانية.

وحدات قياس الطول هي الوحدات المستخدمة لقياس المسافة والطول، ومن هذه الوحدات: الكيلومتر (كم)، المتر (م)،الديسيمتر (دسم)، السنتمتر (سم)، المليمتر (ملم)، ويُرجى مُلاحظة أن ترتيب الوحدات السابقة بشكلٍ تنازلي.

وفيما يأتي بيان لعلاقة كل منها بالأخرى:

كل واحد متر= 10 دسم.

كل واحد متر= 100سم.

كل واحد متر=1000ملم.

كل واحد ديسيمتر= 10سم.

كل واحد سنتمتر= 10 ملم.

وعند تحويل الوحدة من الأكبر إلى الأصغر منها يُستخدم الضرب، أما في حال التحويل من الأصغر للأكبر تُستخدم حينها القسمة، حيث تُقسم أو تُضرب الوحدات بإحدى الأعداد الآتية: 100،10، 1000، 10000. ولفهم هذا الأمر بشكلٍ أبسط يمكن تمثيل هذه الوحدات على شكل سُلم، حيث تُمثل أصغر وحدة الدرجة الأولى، والوحدة الأكبر منها تمثل الدرجة الثانية وهكذا.

أمثلة على تحويل وحدات النقد

مثال: وزّع معلمٌ هدية عينيّةً لكل طالب مقدارها 25 قرشاً، فإذا علمت أن عدد الطلاب الذين وُزّعت عليهم الهدايا هم 80 طالباً، فكم ديناراً تكلفة الهدايا التي تم توزيعها على الطلاب.

الحل:

تكلفة الهدايا الموزعة على الطلاب بالقروش= تكلفة الهدية × عدد الطلاب.

تكلفة الهدايا الموزعة على الطلاب بالقروش= 25 قرشاً × 80 طالب.

تكلفة الهدايا الموزعة على الطلاب بالقروش=2000 قرشاً.

عند التحويل من وحدة صغيرة إلى وحدة أكبرفإننا نقسم.

التكلفة بالدينار= 2000 ÷100= 20 ديناراً.

إذن تكلفة الهدايا التي قدمها المُعلم لطلابه هي 20 ديناراً.

أمثلة على تحويل وحدات الزمن

مثال: بدأ إحدى الموظفين عمله في الساعة السابعة والنصف صباحاً، فإذا كانت ساعات دوامه الرسمية هي 6 ساعات و45 دقيقة، فجد الساعة التي ينهي فيها هذا الموظف دوامه.

الحل:

الساعة السابعة ونصف+ 6 ساعات و45 دقيقة.

7 ساعات و30 دقيقة+ 6ساعات و45 دقيقة

= 13 ساعة و75 دقيقة، (تُجمع الساعات مع الساعات، والدقائق مع الدقائق).

75 دقيقة عبارة عن 1 ساعة+ 15 دقيقة، (حسب العلاقة 1ساعة = 60 دقيقة).

ينتج: 13 ساعة + 1ساعة+ 15 دقيقة= 14 ساعة و15 دقيقة.

إذن ينهي الموظف دوامه في الساعة الثانية و15 دقيقة بعد الظهر، (الساعة 14 هي الساعة الثانية بعد الظهر).

أمثلة على تحويل وحدات الطول

مثال1: إذا علمت أن المسافة التي قطعها إحدى الأشخاص هي 3 كم، فجد هذه المسافة بالأمتار.

الحل:

نلاحظ أن التحويل المطلوب من الوحدة الكبيرة إلى الصغيرة، وحسب العلاقة، فعند التحويل من (كم) إلى(متر) يتم الضرب بالعدد 1000.

3 كم= 3 × 1000.

3 كم= 3000م.

مثال2: اشترى مهندس 10 قطع من الأسلاك، طول كل قطعة 20 سم، جِد طول الأسلاك التي اشتراها بالأمتار.

الحل:

في البداية علينا إيجاد طول الأسلاك بالسنتمترات، طول السلك بالسنتمترات= حاصل ضرب طول القطعة الواحدة بعدد القطع.

طول الأسلاك بالسنتمترات= 20 × 10= 200 سم.

حسب العلاقة، فإن كل واحد مترٍ يساوي 100 سنتمتر، وعند التحويل من (سم) إلى (م) يُقسم على 100.

طول الأسلاك بالأمتار= 200 ÷ 100.

إذن: طول الأسلاك= 2 م.

فيديو عن وحدات قياس المسافة

للتعرف على المزيد حول وحدات قياس المسافة شاهد الفيديو.

1. محمد ابراهيم خشان (2009)، ^ مناهج الرياضيات وأساليب تدريها للصفوفف الأساسية الطبعة الأولى، عمان- الأردن: دار الجنادرية للنشر والتوزيع، صفحة 263-267، الجزء الأول. بتصرّف.

2. باجس خمايسة، ابراهيم الصماي، فدوى الحشاش (2007)، ^ دليل المعلم الرياضيات الصف الخامس (الطبعة الأولى)، الأردن-عمان: وزارة التربية والتعليم إدارة المناهج والكتب المدرسية، صفحة 94-100، جزء الأول. بتصرّف.

3. خالد الرواضية (2016)، ^ رفيقُ الأزماتِ لمعالجة الضعف في الرياضياتِ (الطبعة الأولى)، الأردن-عمان: دار العنقاء، صفحة 61-66، جزء الأول. بتصرّف.

# data[].attributes.attachment_link.title
ما هي وحدات القياس - موضوع

# data[].attributes.attachment_link.description
القياس: هي مُقارنة بين شيئين أو أكثر، كمقارنة طول شخصين، حيث يُقاس طول كل شخص باستخدام الأداة المناسبة، ليُبنى بعدها الحُكم حول أيهما أطول من الآخر. أما وحدات القياس: فهي تُحوّل ما تم مقارنته إلى كميات معروفة، ومن أمثلة الكميات المُراد قياسها ما يأتي: النقد، والوقت، والطول والكتلة، والقوة، والحجم، والحرارة، وأيضاً الضغط، والطاقة.

# data[].attributes.headline
التدريبات الإثرائية لباب حساب المثلثات

# data[].attributes.subject
التدريبات الإثرائية لباب حساب المثلثات

# data[].attributes.attachment_link.title
حساب المثلثات

# data[].attributes.headline
التدريبات الإثرائية لباب الاحتمالات

# data[].attributes.subject
التدريبات الإثرائية لباب الاحتمالات

# data[].attributes.headline
التدريبا الإثرائية لباب المتتابعات والمتسلسلات

# data[].attributes.subject
التدريبا الإثرائية لباب المتتابعات والمتسلسلات

# data[].attributes.attachment_link.title
المتابعات والمتسلسلات

# data[].attributes.headline
التدريبات الاثرائية لباب العلاقات والدوال النسبية

# data[].attributes.subject
التدريبات الاثرائية لباب العلاقات والدوال النسبية

# data[].attributes.attachment_link.title
العلاقات والدوال النسبية

# data[].attributes.attachment_link.title
المهارات الناجحة التي تتبعها في الحياة

# data[].attributes.attachment_link.title
حل مسائل المصفوفات بطريقة مميزة

# data[].attributes.attachment_link.title
التعلم النشط

# data[].attributes.headline
العلاقة بين الرياضيات والفيزياء

# data[].attributes.subject
العلاقة بين الرياضيات والفيزياء

# data[].attributes.attachment_link.authors[].name
العلوم الفيزيائية

# data[].attributes.attachment_link.title
ماهي العلاقة بين الرياضيات والفيزياء ؟ العالم ميتشو كاكو *مترجم*

# data[].attributes.headline
دعاء سعيد الشرقاوي

# data[].attributes.subject
دعاء سعيد الشرقاوي

# data[].attributes.body
كيف يساعدنا التفكير الرياضي على فهم الأمور في حياتنا؟

# data[].attributes.attachment_link.title
Mathematical Thinking: Crash Course Statistics #2

# data[].attributes.attachment_link.authors[].name
Mathsaudi رياضيات السعودية

# data[].attributes.attachment_link.title
#رياضيات 3 - دوال خاصة - شرح الدرس

# data[].attributes.headline
ارجوان فطاني

# data[].attributes.subject
ارجوان فطاني

# data[].attributes.body
الدوال الحسابية هي المواد الأساسية في دراسة النظرية الحسابية . الدوال الحسابية هي التماثلية الرسمية للفكرة البديهية للخوارزمية. . وهي تستخدم لمناقشة الحسابية دون الإشارة إلى أي نموذج ملموس من الحساب

# data[].attributes.headline
حقائق مدهشة ربما لم ت

# data[].attributes.body
حقائق مدهشة ربما لم تلاحظها عن الرياضيات

تحتل الأرقام مكاناً كبيراً من حياتنا اليومية، إذ ترتبط بكل ما حولنا، وعلى الرغم من أن البعض يجدها مخيفة في بعض الأحيان، إلا أنها ممتعة ومذهلة في الوقت ذاته، ولذلك نتطرق الآن إلى بعض الحقائق المثيرة للاهتمام الذي تتعلق بالرياضيات.

1- المائة تعني في الواقع مائة وعشرون

اشتقت كلمة "hundred" التي تعني مائة من كلمة "hundrath" والتي تعني مائة وعشرون.

2- جوجل مشتقة من الرياضيات

اشتقت كلمة محرك البحث جوجل "Google" من كلمة جوجول "Googol" والتي تعني عشرة مرفوعة لأس 100.

3- جنون المضاعفات

هناك بعض المضاعفات تسير بطرق مجنونة؛ فعلى سبيل المثال العدد 111×111، يعطي ناتج 12321، فتستطيع قراءته من الجانبين  وكذلك 1111×1111، وكذلك عند الزيادة.

4- شكسبير والرياضيات

يعتبر شكسبير أحد أعظم الأدباء، وما زالت أعماله موجودة حتى الآن، وبالرغم من كثرة أعماله إلا أنها لم تأتي بها كلمة الرياضيات "Mathematics" إلا مرة واحدة في مسرحية "The Taming of the Shrew".

5- "Math" أم "Mathematics"

أطلق الأمريكيون كلمة " math" على الرياضيات، في حين كان يطلق عليها حينها "mathematics" وعللوا ذلك بأنهم يفضلون هذا الاسم، ويحمل نفس المعنى.

6- الصفر في الرومانية

لم يكن موجوداً في الأرقام الرومانية، ولكنه اشتق من الكلمة العربية "سيفر"  والتي تعني لا شيء.

7- سحر العدد تسعة

عند ضرب أي عدد مع الرقم تسعة، فدائماً ما يكون مجموع أرقام العدد الناتج مساوياً للرقم تسعة. 9×5=45 وعند جمع الأرقام التي يتكون منها الناتج نجد 4+5=9.

8- الصفر عدد زوجي

تم الاتفاق أن الصفر يتبع الأعداد الزوجية، ومن المعروف أن مجموع عدد زوجي مع آخر فردي ينتج عنه عدد فردي لا محالة، ولكن عند جمع الصفر على أي عدد زوجي ينتج عدد زوجي؛ لذا اعتبره العلماء  من الأعداد الزوجية.

9- التفاضل والتكامل

في اللغة اليونانية، تعتبر كلمة " Pebbles" هي المرادف لكلمة " Calculus" التي تعني التفاضل والتكامل، ولكن معناها في اليونانية الحصى، وسبب التسمية لأن عالم الرياضيات اليوناني فيثاغورس كان يستخدم الصخور الصغيرة للدلالة على الأرقام أثناء العمل على المعادلات الرياضية.

10- البيتزا ورمز الباي π
تأخذ البيتزا الشكل الدائري، ولها ارتفاع أيضاً، فإذا اعتبرنا نصف قطر الدائرة " z"، والارتفاع " a"، فإن الحجم يكون ط نق²×ع أي تساوي pi×z×z×a وعند جمع الحروف تصبح pizza.