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.

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Let *a* = {*a*_{i}: i ≥ 0} be any real-valued sequence with *a*_{i} ≥ 0 for all *i*, and let

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.

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*.

- Abelian theorem at PlanetMath; a more general look at Abelian theorems of this type.