Abel's theorem

Abel's theorem This article is about Abel's theorem on power series. For Abel's theorem on algebraic curves, see Abel–Jacobi map. For Abel's theorem on the insolubility of the quintic equation, see Abel–Ruffini theorem. For Abel's theorem on linear differential equations, see Abel's identity. For Abel's theorem on irreducible polynomials, see Abel's irreducibility theorem. For Abel's formula for summation of a series, using an integral, see Abel's summation formula. This article includes a list of references, leitura relacionada ou links externos, mas suas fontes permanecem obscuras porque faltam citações em linha. Ajude a melhorar este artigo introduzindo citações mais precisas. (Fevereiro 2013) (Saiba como e quando remover esta mensagem de modelo) Na matemática, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel.

Conteúdo 1 Teorema 2 Observações 3 Formulários 4 Esquema da prova 5 Related concepts 6 Veja também 7 Leitura adicional 8 External links Theorem Let the Taylor series {estilo de exibição G(x)=soma _{k=0}^{infty }uma_{k}x^{k}} be a power series with real coefficients {estilo de exibição a_{k}} with radius of convergence {estilo de exibição 1.} Suppose that the series {soma de estilo de exibição _{k=0}^{infty }uma_{k}} converge. Então {estilo de exibição G(x)} is continuous from the left at {displaystyle x=1,} isso é, {displaystyle lim _{xto 1^{-}}G(x)=soma _{k=0}^{infty }uma_{k}.} The same theorem holds for complex power series {estilo de exibição G(z)=soma _{k=0}^{infty }uma_{k}z^{k},} provided that {displaystyle zto 1} entirely within a single Stolz sector, isso é, a region of the open unit disk where {estilo de exibição |1-z|leq M(1-|z|)} for some fixed finite {displaystyle M>1} . Without this restriction, the limit may fail to exist: por exemplo, the power series {soma de estilo de exibição _{n>0}{fratura {z^{3^{n}}-z^{2cdot 3^{n}}}{n}}} converge para {estilo de exibição 0} no {estilo de exibição z = 1,} but is unbounded near any point of the form {estilo de exibição e^{pi i/3^{n}},} so the value at {estilo de exibição z = 1} is not the limit as {estilo de exibição com} tends to 1 in the whole open disk.

Observe que {estilo de exibição G(z)} is continuous on the real closed interval {estilo de exibição [0,t]} por {estilo de exibição t<1,} by virtue of the uniform convergence of the series on compact subsets of the disk of convergence. Abel's theorem allows us to say more, namely that {displaystyle G(z)} is continuous on {displaystyle [0,1].} Remarks As an immediate consequence of this theorem, if {displaystyle z} is any nonzero complex number for which the series {displaystyle sum _{k=0}^{infty }a_{k}z^{k}} converges, then it follows that {displaystyle lim _{tto 1^{-}}G(tz)=sum _{k=0}^{infty }a_{k}z^{k}} in which the limit is taken from below. The theorem can also be generalized to account for sums which diverge to infinity.[citation needed] If {displaystyle sum _{k=0}^{infty }a_{k}=infty } then {displaystyle lim _{zto 1^{-}}G(z)to infty .} However, if the series is only known to be divergent, but for reasons other than diverging to infinity, then the claim of the theorem may fail: take, for example, the power series for {displaystyle {frac {1}{1+z}}.} At {displaystyle z=1} the series is equal to {displaystyle 1-1+1-1+cdots ,} but {displaystyle {tfrac {1}{1+1}}={tfrac {1}{2}}.} We also remark the theorem holds for radii of convergence other than {displaystyle R=1} : let {displaystyle G(x)=sum _{k=0}^{infty }a_{k}x^{k}} be a power series with radius of convergence {displaystyle R,} and suppose the series converges at {displaystyle x=R.} Then {displaystyle G(x)} is continuous from the left at {displaystyle x=R,} that is, {displaystyle lim _{xto R^{-}}G(x)=G(R).} Applications The utility of Abel's theorem is that it allows us to find the limit of a power series as its argument (that is, {displaystyle z} ) approaches {displaystyle 1} from below, even in cases where the radius of convergence, {displaystyle R,} of the power series is equal to {displaystyle 1} and we cannot be sure whether the limit should be finite or not. See for example, the binomial series. Abel's theorem allows us to evaluate many series in closed form. For example, when {displaystyle a_{k}={frac {(-1)^{k}}{k+1}},} we obtain {displaystyle G_{a}(z)={frac {ln(1+z)}{z}},qquad 0by integrating the uniformly convergent geometric power series term by term on {estilo de exibição [-z,0]} ; thus the series {soma de estilo de exibição _{k=0}^{infty }{fratura {(-1)^{k}}{k+1}}} converge para {estilo de exibição ln(2)} by Abel's theorem. De forma similar, {soma de estilo de exibição _{k=0}^{infty }{fratura {(-1)^{k}}{2k+1}}} converge para {displaystyle arctan(1)={tfrac {pi }{4}}.} {estilo de exibição G_{uma}(z)} is called the generating function of the sequence {displaystyle a.} Abel's theorem is frequently useful in dealing with generating functions of real-valued and non-negative sequences, such as probability-generating functions. Em particular, it is useful in the theory of Galton–Watson processes. Outline of proof After subtracting a constant from {estilo de exibição a_{0},} we may assume that {soma de estilo de exibição _{k=0}^{infty }uma_{k}=0.} Deixar {estilo de exibição s_{n}=soma _{k=0}^{n}uma_{k}!.} Then substituting {estilo de exibição a_{k}=s_{k}-s_{k-1}} and performing a simple manipulation of the series (summation by parts) resulta em {estilo de exibição G_{uma}(z)=(1-z)soma _{k=0}^{infty }s_{k}z^{k}.} Given {displaystyle varepsilon >0,} pick {estilo de exibição m} large enough so that {estilo de exibição |s_{k}|

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