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, letture correlate o collegamenti esterni, ma le sue fonti rimangono poco chiare perché mancano di citazioni inline. Aiutaci a migliorare questo articolo introducendo citazioni più precise. (febbraio 2013) (Scopri come e quando rimuovere questo messaggio modello) In matematica, 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.

Contenuti 1 Teorema 2 Osservazioni 3 Applicazioni 4 Schema di prova 5 Related concepts 6 Guarda anche 7 Ulteriori letture 8 External links Theorem Let the Taylor series {stile di visualizzazione G(X)=somma _{k=0}^{infty }un_{K}x^{K}} be a power series with real coefficients {stile di visualizzazione a_{K}} with radius of convergence {stile di visualizzazione 1.} Suppose that the series {somma dello stile di visualizzazione _{k=0}^{infty }un_{K}} converge. Quindi {stile di visualizzazione G(X)} is continuous from the left at {displaystyle x=1,} questo è, {displaystyle lim _{xto 1^{-}}G(X)=somma _{k=0}^{infty }un_{K}.} The same theorem holds for complex power series {stile di visualizzazione G(z)=somma _{k=0}^{infty }un_{K}z^{K},} provided that {displaystyle zto 1} entirely within a single Stolz sector, questo è, a region of the open unit disk where {stile di visualizzazione |1-z|leq M(1-|z|)} for some fixed finite {displaystyle M>1} . Without this restriction, the limit may fail to exist: Per esempio, the power series {somma dello stile di visualizzazione _{n>0}{frac {z^{3^{n}}-z^{2cdot 3^{n}}}{n}}} converge a {stile di visualizzazione 0} a {stile di visualizzazione z = 1,} but is unbounded near any point of the form {stile di visualizzazione e^{pi i/3^{n}},} so the value at {stile di visualizzazione z = 1} is not the limit as {stile di visualizzazione con} tends to 1 in the whole open disk.

Notare che {stile di visualizzazione G(z)} is continuous on the real closed interval {stile di visualizzazione [0,t]} per {stile di visualizzazione 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 {stile di visualizzazione [-z,0]} ; thus the series {somma dello stile di visualizzazione _{k=0}^{infty }{frac {(-1)^{K}}{k+1}}} converge a {stile di visualizzazione ln(2)} by Abel's theorem. Allo stesso modo, {somma dello stile di visualizzazione _{k=0}^{infty }{frac {(-1)^{K}}{2k+1}}} converge a {displaystyle arctan(1)={tfrac {pi }{4}}.} {stile di visualizzazione G_{un}(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. In particolare, it is useful in the theory of Galton–Watson processes. Outline of proof After subtracting a constant from {stile di visualizzazione a_{0},} we may assume that {somma dello stile di visualizzazione _{k=0}^{infty }un_{K}=0.} Permettere {stile di visualizzazione s_{n}=somma _{k=0}^{n}un_{K}!.} Then substituting {stile di visualizzazione a_{K}=s_{K}-S_{k-1}} and performing a simple manipulation of the series (summation by parts) risulta in {stile di visualizzazione G_{un}(z)=(1-z)somma _{k=0}^{infty }S_{K}z^{K}.} Given {displaystyle varepsilon >0,} pick {stile di visualizzazione n} large enough so that {stile di visualizzazione |S_{K}|

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