사용자:Adrenalin/작업장 5

급수란 수학에서 수열들의 각 항의 합을 의미한다. 즉, 급수란 여러 수들의 합연산으로 표현된다. 급수의 예로는 아래와 같은 등차수열의 합이 있다.

1 + 2 + 3 + 4 + 5 + ... + 99 + 100

급수에 더해지는 각 항들이 어떤 공식이나 알고리즘에 의해 표현되는 경우도 있다. 난수들로 이루어진 급수도 생각할 수 있다.

급수는 유한 급수와 무한 급수로 나눌 수 있다. 유한 급수의 경우 기초적인 대수학의 법칙들만 사용하여도 그 값을 구할 수 있다. 하지만 무한 급수는 그 정확한 합을 구하기 위해서는 해석학의 여러 정리들이 필요하다. 예를 들어 등차수열들의 합으로 이루어진 급수의 경우, 다음과 같이 나타낼 수 있다.

$\sum _{n=0}^{k}(an+b);$

등비수열의 합으로 이루어진 급수의 경우, 다음과 같이 나타낼 수 있다.

$\sum _{n=0}^{k}a^{n}.$

무한급수

$S_{n}=a_{0}+a_{1}+a_{2}+\cdots +a_{n},$

무한급수의 합은 위에 적혀있는, 급수의 부분합으로 이루어지는 수열의 극한값으로 생각한다. n이 무한대로 갈 때 그 극한이 유한한 값을 갖는다면 이 급수가 수렴한다고 한다. 만약 이 값이 무한하거나 존재하지 않는다면, 이 급수는 발산한다고 한다.

무한급수가 발산하는지 여부를 판단하는 가장 쉬운 방법은 급수를 구성하고 있는 수열의 n번째 항인 a_n이 n이 무한으로 갈 때 0으로 수렴하는지 여부를 체크하면 된다. 만약 0으로 가지 않는다면, 이 급수는 발산한다는 사실을 쉽게 확인할 수 있다. 하지만 그 극한값이 0으로 간다고 해도, 이 급수가 항상 수렴하는 것은 아니다. 다음의 급수의 경우 수열의 값은 0으로 수렴하지만, 급수는 수렴하지 않는다.

$1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots .$

급수를 구성하고 있는 각 수열들이 0이 아닌 항으로만 이루어져 있더라도 수렴할 수도 있다. 제논의 역설로도 확인할 수 있는 수렴하는 무한급수의 예는 다음과 같다.

${\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots .$

수직선에서 이를 눈으로 확인해볼 수 있다. 수직선에서 ${\frac {1}{2}},{\frac {3}{4}},{\frac {7}{8}},\cdots$에 해당하는 부분에 점을 찍어보면, 언제나 마지막에 찍은 점과 그 앞에 찍은 점 사이의 거리가, 1과 마지막에 찍은 점과의 거리와 같다는 사실을 알 수 있다. 하지만 이런 논리로는 이 급수의 부분합이 항상 1보다 작다는 사실을 설명할 뿐, 무한급수의 합이 1이 된다는 사실을 증명해주지는 못한다.

위의 급수를 등비급수로 표현하면 다음과 같이 쓸 수 있다.

$\sum _{n=0}^{\infty }2^{-n+1}=2.$

일반적인 무한급수를 표시할 때에는 다음과 같이 쓸 수 있다.

$\sum _{n=0}^{\infty }a_{n}$

여기에서 a_n은 실수(혹은 복소수)이며, 만약 부분합의 극한인

$\lim _{N\rightarrow \infty }\sum _{n=0}^{N}a_{n}$

이 어떤 값 S로 수렴한다면, 이 무한급수의 합은 S와 같다고 한다. 이런 수 S가 존재하지 않을 경우 이 급수는 발산한다고 한다.

정의

수학자들은 흔히 급수를 다룰 때, 급수들의 각 항을 구성하는 수열 a₀, a₁, a₂, …과 그 수열의 부분합으로 이루어진 수열 S₀, S₁, S₂, …을 함께 생각한다. 이 때, S_n = a₀ + a₁ + … + a_n가 성립한다. 다음과 같은 수식 표현은 이 급수의 정의, 수렴 여부에 관계 없이 해당 수열의 부분합들의 극한을 의미한다. 수렴하는 경우, 이 수식 표현은 또한 부분합으로 이루어진 수열의 극한값을 의미한다.

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, functions, 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 9 × 0.111… = 0.999… = 1, 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 fractions. He discovered a number of infinite series, including the Taylor series of the trigonometric functions 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

$1+{\frac {\alpha \beta }{1\cdot \gamma }}x+{\frac {\alpha (\alpha +1)\beta (\beta +1)}{1\cdot 2\cdot \gamma (\gamma +1)}}x^{2}+\cdots .$

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

$1+{\frac {m}{1}}x+{\frac {m(m-1)}{2!}}x^{2}+\cdots$

corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values of $m$ and $x$. 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 (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

$F(x)=1^{n}+2^{n}+\cdots +(x-1)^{n}.\,$

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

• A geometric series is one where each successive term is produced by multiplying the previous term by a constant number. Example:

$1+{1 \over 2}+{1 \over 4}+{1 \over 8}+{1 \over 16}+\cdots =\sum _{n=0}^{\infty }{1 \over 2^{n}}.$

In general, the geometric series

$\sum _{n=0}^{\infty }z^{n}$

converges if and only if |z| < 1.

• The harmonic series is the series

$1+{1 \over 2}+{1 \over 3}+{1 \over 4}+{1 \over 5}+\cdots =\sum _{n=1}^{\infty }{1 \over n}.$

The harmonic series is divergent.

• An alternating series is a series where terms alternate signs. Example:

$1-{1 \over 2}+{1 \over 3}-{1 \over 4}+{1 \over 5}-\cdots =\sum _{n=1}^{\infty }(-1)^{n+1}{1 \over n}.$

• The series

$\sum _{n=1}^{\infty }{\frac {1}{n^{r}}}$

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

• A telescoping series

$\sum _{n=1}^{\infty }(b_{n}-b_{n+1})$

converges if the sequence b_n converges to a limit L as n goes to infinity. The value of the series is then b₁ − L.

Absolute convergence

 이 부분의 본문은 absolute convergence입니다.

A series

$\sum _{n=0}^{\infty }a_{n}$

is said to converge absolutely if the series of absolute values

$\sum _{n=0}^{\infty }\left|a_{n}\right|$

converges. In this case, the original series, and all reorderings of it, converge, and converge towards the same sum.

The Riemann series theorem says that if a series is conditionally convergent then one can always find a reordering of the terms so that the reordered series diverges. Moreover, if the a_n are real and S is any real number, one can find a reordering so that the reordered series converges with limit S.

Convergence tests

 이 부분의 본문은 convergence tests입니다.

• Comparison test 1: If ∑b_n  is an absolutely convergent series such that |a_n | ≤ C |b_n | for some number C  and for sufficiently large n , then ∑a_n  converges absolutely as well. If ∑|b_n | diverges, and |a_n | ≥ |b_n | for all sufficiently large n , then ∑a_n  also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the a_n  alternate in sign).
• Comparison test 2: If ∑b_n  is an absolutely convergent series such that |a_n+1 /a_n | ≤ |b_n+1 /b_n | for sufficiently large n , then ∑a_n  converges absolutely as well. If ∑|b_n | diverges, and |a_n+1 /a_n | ≥ |b_n+1 /b_n | for all sufficiently large n , then ∑a_n  also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the a_n  alternate in sign).
• Ratio test: If |a_n+1/a_n| approaches a number less than one as n approaches infinity, then ∑ a_n converges absolutely. When the ratio is 1, convergence can sometimes be determined as well.
• Root test: If there exists a constant C < 1 such that |a_n|^1/n ≤ C for all sufficiently large n, then ∑ a_n converges absolutely.
• Integral test: if f(x) is a positive monotone decreasing function defined on the interval [1, ∞) with f(n) = a_n for all n, then ∑ a_n converges if and only if the integral ∫₁^∞ f(x) dx is finite.
• Alternating series test: A series of the form ∑ (−1)ⁿ a_n (with a_n ≥ 0) is called alternating. Such a series converges if the sequence a_n is monotone decreasing and converges to 0. The converse is in general not true.
• n-th term test: If lim_n→∞ a _n ≠ 0 then the series diverges.
• For some specific types of series there are more specialized convergence tests, for instance for Fourier series there is the Dini test.

Power series

Several important functions can be represented as Taylor series; these are infinite series involving powers of the independent variable and are also called power series. For example, the series

$\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}$

converges to $e^{x}$ for all x. See also radius of convergence.

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. However, the formal operation with non-convergent series has been retained in rings of formal power series which are studied in abstract algebra. Formal power series are also used in combinatorics to describe and study sequences that are otherwise difficult to handle; this is the method of generating functions.

Dirichlet series

 이 부분의 본문은 Dirichlet series입니다.

A Dirichlet series is one of the form

$\sum _{n=1}^{\infty }{a_{n} \over n^{s}},$

where s is a complex number. Generally these converge if the real part of s is greater than a number called the abscissa of convergence.

Generalizations

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.

Cesàro summation, (C,k) summation, Abel summation, and Borel summation provide increasingly weaker (and hence applicable to increasingly divergent series) means of defining the sums of series.

The notion of series can be defined in every abelian topological group; the most commonly encountered case is that of series in a Banach space.

Summations over arbitrary index sets

Analogous definitions may be given for sums over arbitrary index set. Let a: I → X, where I is any set and X is an abelian topological group. Let F be the collection of all finite subsets of I. Note that F is a directed set ordered under inclusion with union as join. We define the sum of the series as the limit

$\sum _{i\in I}a_{i}=\lim _{F}\left\{\sum _{i\in A}a_{i}\,{\bigg |}A\in F\right\}$

if it exists and say that the series a converges unconditionally. Thus it is the limit of all finite partial sums. Because F is not totally ordered, and because there may be uncountably many finite partial sums, this is not a limit of a sequence of partial sums, but rather of a net.

Note, however that $\scriptstyle \{i\in I:a_{i}>0\}$ needs to be countable for the sum to be finite. To see this, suppose it is uncountable. Then some $\scriptstyle A_{n}=\{i\in I:a_{i}>1/n\}$ would be uncountable, and we can estimate the sum

$\sum _{i\in I}a_{i}\geq {\textrm {card}}(A_{n})/n=\infty .$

This definition is insensitive to the order of the summation, so the limit will not exist for conditionally convergent series. If, however, I is a well-ordered set (for example any ordinal), one may consider the limit of partial sums of the finite initial segments

$\sum _{i\in I}a_{i}=\lim _{n\to \infty }\sum _{i=1}^{n}a_{i}.$

If this limit exists, then the series converges. Unconditional convergence implies convergence, but not conversely, as in the case of real sequences. If X is a Banach space and I is well-ordered, then one may define the notion of absolute convergence. A series converges absolutely if

$\lim _{n\to \infty }\sum _{i=1}^{n}\|a_{i}\|$

exists. If a sequence converges absolutely then it converges unconditionally, but the converse only holds in finite dimensional Banach spaces.

Note that in some cases if the series is valued in a space that is not separable, one should consider limits of nets of partial sums over subsets of I which are not finite.

Real sequences

For real-valued series, an uncountable sum converges only if at most countably many terms are nonzero. Indeed, let

$I_{n}=\left\{i\in I\,{\bigg |}a_{i}>{\frac {1}{n}}\right\}$

be the set of indices whose terms are greater than 1/n. Each I_n is finite (otherwise the series would diverge). The set of indices whose terms are nonzero is the union of the I_n by the Archimedean principle, and the union of countably many countable sets is countable by the axiom of choice.

Occasionally integrals of real functions are described as sums over the reals. The above result shows that this interpretation should not be taken too literally. On the other hand, any sum over the reals can be understood as an integral with respect to the counting measure, which accounts for the many similarities between the two constructions.

The proof goes forward in general first-countable topological vector spaces as well, such as Banach spaces; define I_n to be those indices whose terms are outside the n-th neighborhood of 0. Thus uncountable series can only be interesting if they are valued in spaces that are not first-countable.

Examples

1. Given a function f: X→Y, with Y an abelian topological group, then define

   $f_{a}(x)={\begin{cases}0&x\neq a,\\f(a)&x=a,\\\end{cases}}$

   the function whose support is a singleton {a}. Then

   $f=\sum _{a\in X}f_{a}$

   in the topology of pointwise convergence. This space is separable but not first countable.
2. On the first uncountable ordinal viewed as a topological space in the order topology, the constant function f: [0,ω₁] → [0,ω₁] given by f(α) = 1 satisfies

   $\sum _{\alpha \in [0,\omega _{1}]}f(\alpha )=\omega _{1}$

   (in other words, ω₁ copies of 1 is ω₁) only if one takes a limit over all countable partial sums, rather than finite partial sums. This space is not separable.
3. In the definition of partitions of unity, one constructs sums over arbitrary index. While, formally, this requires a notion of sums of uncountable series, by construction there are only finitely many nonzero terms in the sum, so issues regarding convergence of such sums do not arise.

See also

• Convergent series
• Divergent series
• Sequence transformations
• Infinite product

References

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

External links

• Graphical simulation of series convergence