Abel's theorem

From Wikipedia, the free encyclopedia

For Abel's theorem on the insolubility of the quintic equation with radicals, see Abel-Ruffini theorem.

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

Let a = {ai: i ≥ 0} be any sequence of real or complex numbers and let

G_a(z) = \sum_{i=0}^{\infty} a_i z^i\,

be the power series with coefficients a. Suppose that the series \sum_{i=0}^\infty a_i converges. Then,

\lim_{z\rightarrow 1^-} G_a(z) = \sum_{i=0}^{\infty} a_i.\qquad (*)

In the special case where all the coefficients ai are real and ai ≥ 0 for all i, then the above formula ( * ) holds also when the series \sum_{i=0}^\infty a_i does not converge. I.e. in that case both sides of the formula equal +∞.

In a more general version of this theorem, if r is any nonzero real number for which the series \sum_{i=0}^\infty a_i r^i converges, then it follows that

\lim_{z\to r} G_a(z) = \sum_{i=0}^{\infty} a_ir^i. \, \ \

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.

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.

Ga(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-negative sequences, such as probability-generating functions. In particular, it is useful in the theory of Galton-Watson processes.

Converses to a theorem like Abel's are called Tauberian theorems: there is no exact converse, but results conditional on some hypothesis. The field of divergent series, and their summation methods, contains many theorems of abelian type and of tauberian type.

Retrieved from "http://en.wikipedia.org../../../a/b/e/Abel%27s_theorem.html"
Advanced Search
Included Web Search Engines


Safe Search

close

Top Matching Results

Occasionally Search.com will highlight specialized results that are based on the context of your query. Examples of specialized results include specific links to news, images, or video.

Top Matching Results may highlight information from other Search.com pages, content from the CNET Network of sites, or third party content. The listings are based purely on relevance. Search.com does not receive payment for listings in this section but our partners that provide this data may get paid for listing these products.

Sponsored Links

This section contains paid listings which have been purchased by companies that want to have their sites appear for specific search terms and related content. These listings are administered, sorted and maintained by a third party and are not endorsed by Search.com.

Search Results

Search.com sends your search query to several search engines at one time and integrates the results into one list which has been sorted by relevance using Search.com's proprietary algorithm. You can customize the list of search engines included in your metasearch from the preferences.

The search engines that are used in your metasearch may allow companies to pay to have their Web sites included within the results. To view the Paid Inclusion policy for a specific search engine, please visit their Web site. Search.com does not accept payment or share revenue with any search engine partner for listings in this section.