Abel's theorem

In real analysis, '''Abel's theorem''' for power series with non-negative coefficients relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel. For Abel's theorem on algebraic curves, see Jacobian variety. ==Theorem== Let ''a'' = {''a''_''i'': ''i'' ≥ 0} be any real-valued sequence with ''a''_''i'' ≥ 0 for all ''i'', and let :

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

be the power series with coefficients ''a''. Then, :

\lim_{z\uparrow 1} G_a(z) = \sum_{i=0}^{\infty} a_i,\,

whether or not this sum is finite. ==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. ''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-negative sequences, such as probability-generating functions. In particular, it is useful in the theory of Galton-Watson processes. ==Related concepts== 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''. ==External links== * [http://planetmath.org/encyclopedia/AbelianTheorem.html Abelian theorem] at [http://www.planetmath.org/ PlanetMath]; a more general look at Abelian theorems of this type. Mathematical analysis Real analysisTheoremsMathematical series

See other meanings of words starting from letter:

A

AB | AC | AD | AE | AF | AG | AH | AI | AJ | AK | AL | AM | AN | AO | AP | AR | AS | AT | AU | AW | AX | AY | AZ |

Words begining with Abel\'s_theorem:

Abel's_theorem