사용자:Hwangjy9/작업장3

수학에서, 발산급수는 수렴하지 않는 무한급수를 말한다.

급수가 수렴한다면, 급수의 각 항들은 0으로 접근한다. 그러므로 각 항이 0으로 접근하지 않는 모든 급수는 발산한다. 하지만, 수렴하는 것에 대한 조건이 더 강력한데, 각 항이 0으로 수렴하는 모든 급수가 수렴하는 것은 아니다. 가장 간단한 반례는 조화급수이다.

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

조화급수가 발산한다는 것은 중세 수학자 Nicole Oresme에 의해 증명되었다.

In specialized mathematical contexts, values can be usefully assigned to certain series whose sequence of partial sums diverges. A summability method or summation method is a partial function from the set of sequences of partial sums of series to values. For example, Cesàro summation assigns Grandi's divergent series

$1-1+1-1+\cdots$

the value ¹/₂. Cesàro summation is an averaging method, in that it relies on the arithmetic mean of the sequence of partial sums. Other methods involve analytic continuations of related series. In physics, there are a wide variety of summability methods; these are discussed in greater detail in the article on regularization.

발산급수를 더하는 방법에 대한 정리

A summability method M is regular if it agrees with the actual limit on all convergent series. Such a result is called an abelian theorem for M, from the prototypical Abel's theorem. More interesting and in general more subtle are partial converse results, called tauberian theorems, from a prototype proved by Alfred Tauber. Here partial converse means that if M sums the series Σ, and some side-condition holds, then Σ was convergent in the first place; without any side condition such a result would say that M only summed convergent series (making it useless as a summation method for divergent series).

The operator giving the sum of a convergent series is linear, and it follows from the Hahn–Banach theorem that it may be extended to a summation method summing any series with bounded partial sums. This fact is not very useful in practice since there are many such extensions, inconsistent with each other, and also since proving such operators exist requires invoking the axiom of choice or its equivalents, such as Zorn's lemma. They are therefore nonconstructive.

The subject of divergent series, as a domain of mathematical analysis, is primarily concerned with explicit and natural techniques such as Abel summation, Cesàro summation and Borel summation, and their relationships. The advent of Wiener's tauberian theorem marked an epoch in the subject, introducing unexpected connections to Banach algebra methods in Fourier analysis.

Summation of divergent series is also related to extrapolation methods and sequence transformations as numerical techniques. Examples for such techniques are Padé approximants, Levin-type sequence transformations, and order-dependent mappings related to renormalization techniques for large-order perturbation theory in quantum mechanics.

Properties of summation methods

Summation methods usually concentrate on the sequence of partial sums of the series. While this sequence does not converge, we may often find that when we take an average of larger and larger initial terms of the sequence, the average converges, and we can use this average instead of a limit to evaluate the sum of the series. So in evaluating a = a₀ + a₁ + a₂ + ..., we work with the sequence s, where s₀ = a₀ and s_n+1 = s_n + a_n+1. In the convergent case, the sequence s approaches the limit a. A summation method can be seen as a function from a set of sequences of partial sums to values. If A is any summation method assigning values to a set of sequences, we may mechanically translate this to a series-summation method A^Σ that assigns the same values to the corresponding series. There are certain properties it is desirable for these methods to possess if they are to arrive at values corresponding to limits and sums, respectively.

1. Regularity. A summation method is regular if, whenever the sequence s converges to x, A(s) = x. Equivalently, the corresponding series-summation method evaluates A^Σ(a) = x.
2. Linearity. A is linear if it is a linear functional on the sequences where it is defined, so that A(k r + s) = k A(r) + A(s) for sequences r, s and a real or complex scalar k. Since the terms a_n = s_n+1 − s_n of the series a are linear functionals on the sequence s and vice versa, this is equivalent to A^Σ being a linear functional on the terms of the series.
3. Stability. If s is a sequence starting from s₀ and s′ is the sequence obtained by omitting the first value and subtracting it from the rest, so that s′_n = s_n+1 − s₀, then A(s) is defined if and only if A(s′) is defined, and A(s) = s₀ + A(s′). Equivalently, whenever a′_n = a_n+1 for all n, then A^Σ(a) = a₀ + A^Σ(a′).

The third condition is less important, and some significant methods, such as Borel summation, do not possess it.

One can also give a weaker alternative to the last condition.

1. Finite Re-indexability. If s and s′ are two sequences such that there exists a bijection $f:\mathbb {N} \rightarrow \mathbb {N}$ such that s_i = s′_f(i) for all i, and if there exists some $N\in \mathbb {N}$ such that s_i = s′_i for all i > N, then A(s) = A(s′). (In other words, s′ is the same sequence as s, with only finitely many terms re-indexed.) Note that this is a weaker condition than Stability, because any summation method that exhibits Stability also exhibits Finite Re-indexability, but the converse is not true.

A desirable property for two distinct summation methods A and B to share is consistency: A and B are consistent if for every sequence s to which both assign a value, A(s) = B(s). If two methods are consistent, and one sums more series than the other, the one summing more series is stronger.

There are powerful numerical summation methods that are neither regular nor linear, for instance nonlinear sequence transformations like Levin-type sequence transformations and Padé approximants, as well as the order-dependent mappings of perturbative series based on renormalization techniques.

공리적 방법

Taking regularity, linearity and stability as axioms, it is possible to sum many divergent series by elementary algebraic manipulations. For instance, whenever r ≠ 1, the geometric series

${\begin{aligned}G(r,c)&=\sum _{k=0}^{\infty }cr^{k}&&\\&=c+\sum _{k=0}^{\infty }cr^{k+1}&&{\mbox{ (stability) }}\\&=c+r\sum _{k=0}^{\infty }cr^{k}&&{\mbox{ (linearity) }}\\&=c+r\,G(r,c),&&{\mbox{ whence }}\\G(r,c)&={\frac {c}{1-r}},&&\\\end{aligned}}$

can be evaluated regardless of convergence. More rigorously, any summation method that possesses these properties and which assigns a finite value to the geometric series must assign this value. However, when r is a real number larger than 1, the partial sums increase without bound, and averaging methods assign a limit of ∞.

Nörlund means

Suppose p_n is a sequence of positive terms, starting from p₀. Suppose also that

${\frac {p_{n}}{p_{0}+p_{1}+\cdots +p_{n}}}\rightarrow 0.$

If now we transform a sequence s by using p to give weighted means, setting

$t_{m}={\frac {p_{m}s_{0}+p_{m-1}s_{1}+\cdots +p_{0}s_{m}}{p_{0}+p_{1}+\cdots +p_{m}}}$

then the limit of t_n as n goes to infinity is an average called the Nörlund mean N_p(s).

The Nörlund mean is regular, linear, and stable. Moreover, any two Nörlund means are consistent. The most significant of the Nörlund means are the Cesàro sums. Here, if we define the sequence p^k by

$p_{n}^{k}={n+k-1 \choose k-1}$

then the Cesàro sum C_k is defined by C_k(s) = N_(p^k)(s). Cesàro sums are Nörlund means if k ≥ 0, and hence are regular, linear, stable, and consistent. C₀ is ordinary summation, and C₁ is ordinary Cesàro summation. Cesàro sums have the property that if h > k, then C_h is stronger than C_k.

아벨리안 평균

Suppose λ = {λ₀, λ₁, λ₂, ...} is a strictly increasing sequence tending towards infinity, and that λ₀ ≥ 0. Suppose

$f(x)=\sum _{n=0}^{\infty }a_{n}\exp(-\lambda _{n}x)$

converges for all real numbers x>0. Then the Abelian mean A_λ is defined as

$A_{\lambda }(s)=\lim _{x\rightarrow 0^{+}}f(x).$

A series of this type is known as a generalized 디리클레 급수; in applications to physics, this is known as the method of heat-kernel regularization.

Abelian means are regular, linear, and stable, but not always consistent between different choices of λ. However, some special cases are very important summation methods.

아벨 합

 아벨 정리 문서를 참고하십시오.

If λ_n = n, then we obtain the method of Abel summation. Here

$f(x)=\sum _{n=0}^{\infty }a_{n}\exp(-nx)=\sum _{n=0}^{\infty }a_{n}z^{n},$

where z = exp(−x). Then the limit of ƒ(x) as x approaches 0 through positive reals is the limit of the power series for ƒ(z) as z approaches 1 from below through positive reals, and the Abel sum A(s) is defined as

$A(s)=\lim _{z\rightarrow 1^{-}}\sum _{n=0}^{\infty }a_{n}z^{n}$

Abel summation is interesting in part because it is consistent with but more powerful than Cesàro summation: A(s) = C_k(s) whenever the latter is defined. The Abel sum is therefore regular, linear, stable, and consistent with Cesàro summation.

Lindelöf summation

If λ_n = n ln(n), then (indexing from one) we have

$f(x)=a_{1}+a_{2}2^{-2x}+a_{3}3^{-3x}+\cdots .$

Then L(s), the Lindelöf sum (Volkov 2001), is the limit of ƒ(x) as x goes to zero. The Lindelöf sum is a powerful method when applied to power series among other applications, summing power series in the Mittag-Leffler star.

If g(z) is analytic in a disk around zero, and hence has a Maclaurin series G(z) with a positive radius of convergence, then L(G(z)) = g(z) in the Mittag-Leffler star. Moreover, convergence to g(z) is uniform on compact subsets of the star.

See also

• 1 − 1 + 2 − 6 + 24 − 120 + · · ·
• 보렐 합
• Zeta function regularization
• 오일러 합
• Lambert summation
• Silverman–Toeplitz theorem

References

• Arteca, G.A.; Fernández, F.M.; Castro, E.A. (1990), 《Large-Order Perturbation Theory and Summation Methods in Quantum Mechanics》, Berlin: Springer-Verlag .
• Baker, Jr., G. A.; Graves-Morris, P. (1996), 《Padé Approximants》, Cambridge University Press .
• Brezinski, C.; Zaglia, M. Redivo (1991), 《Extrapolation Methods. Theory and Practice》, North-Holland .
• Hardy, G. H. (1949), 《Divergent Series》, Oxford: Clarendon Press .
• LeGuillou, J.-C.; Zinn-Justin, J. (1990), 《Large-Order Behaviour of Perturbation Theory》, Amsterdam: North-Holland .
• Volkov, I.I. (2001). “Lindelöf summation method”. 《Encyclopedia of Mathematics》 (영어). Springer-Verlag. ISBN 978-1-55608-010-4. .
• Zakharov, A.A. (2001). “Abel summation method”. 《Encyclopedia of Mathematics》 (영어). Springer-Verlag. ISBN 978-1-55608-010-4. .