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, related reading or external links, but its sources remain unclear because it lacks inline citations. Please help to improve this article by introducing more precise citations. (February 2013) (Learn how and when to remove this template message) In mathematics, 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.

Contents 1 Theorem 2 Remarks 3 Applications 4 Outline of proof 5 Related concepts 6 See also 7 Further reading 8 External links Theorem Let the Taylor series {displaystyle G(x)=sum _{k=0}^{infty }a_{k}x^{k}} be a power series with real coefficients {displaystyle a_{k}} with radius of convergence {displaystyle 1.} Suppose that the series {displaystyle sum _{k=0}^{infty }a_{k}} converges. Then {displaystyle G(x)} is continuous from the left at {displaystyle x=1,} that is, {displaystyle lim _{xto 1^{-}}G(x)=sum _{k=0}^{infty }a_{k}.} The same theorem holds for complex power series {displaystyle G(z)=sum _{k=0}^{infty }a_{k}z^{k},} provided that {displaystyle zto 1} entirely within a single Stolz sector, that is, a region of the open unit disk where {displaystyle |1-z|leq M(1-|z|)} for some fixed finite {displaystyle M>1} . Without this restriction, the limit may fail to exist: for example, the power series {displaystyle sum _{n>0}{frac {z^{3^{n}}-z^{2cdot 3^{n}}}{n}}} converges to {displaystyle 0} at {displaystyle z=1,} but is unbounded near any point of the form {displaystyle e^{pi i/3^{n}},} so the value at {displaystyle z=1} is not the limit as {displaystyle z} tends to 1 in the whole open disk.

Note that {displaystyle G(z)} is continuous on the real closed interval {displaystyle [0,t]} for {displaystyle 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 00,} pick {displaystyle n} large enough so that {displaystyle |s_{k}|

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