- Abel's theorem
In
mathematics , Abel's theorem forpower series relates a limit of a power series to the sum of itscoefficient s. It is named after Norwegian mathematicianNiels Henrik Abel .Theorem
Let "a" = {"a""k": "k" ≥ 0} be any sequence of real or complex numbersand let
:
be the power series with coefficients "a". Suppose that the series converges. Then
:
In the special case where all the coefficients "a""i" are real and "a""k" ≥ 0 for all "k", then the above formula holds also when the series does not converge. I.e. in that case both sides of the formula equal +∞.
Remark
In a more general version of this theorem, if "r" is any nonzero real number for which the series converges, then it follows that
:
provided we interpret the limit in this formula as a
one-sided limit , from the left if "r" is positive and from the right if "r" is negative.Applications
The utility of Abel's theorem is that it allows us to find the limit of a power series as its argument (i.e. "z") approaches 1 from below, even in cases where the
radius of convergence , "R", of the power series is equal to 1 and we cannot be sure whether the limit should be finite or not. See e.g. thebinomial series ."G""a"("z") is called the
generating function of the sequence "a". Abel's theorem is frequently useful in dealing with generating functions of real-valued and non-negativesequence s, such asprobability-generating function s. In particular, it is useful in the theory ofGalton–Watson process es.Related concepts
Converses to a theorem like Abel's are called
Tauberian theorem s: there is no exact converse, but results conditional on some hypothesis. The field ofdivergent series , and their summation methods, contains many theorems "of abelian type" and "of tauberian type".External links
* "(a more general look at Abelian theorems of this type)"
*
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