View source for Abel criterion

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The term might refer to 
* a criterion for the convergence of series of real numbers
* a related criterion for the convergence of series of complex numbers, used often to determine the convergence of power series at the radius of convergence
* a related criterion for the uniform convergence of a series of functions. 
All these criteria can be proved using [[Summation by parts|summation by parts]], which is also called Abel's lemma or Abel's transformation and it is a discrete version of [[Integration by parts|integration by parts]].

====Criterion for series of real numbers====
If $\sum a_n$ is a convergent series of real numbers and $\{b_n\}$ is a bounded monotone sequence of real numbers, then $\sum_n a_n b_n$ converges.

====Abel test for power series====
Assume that $\{a_n\}$ is a vanishing and monotonically decreasing sequence of real numbers such that the radius of convergence of the power series $\sum_n a_n z^n$ is $1$. Then the series converges at every $z$ with $|z|=1$, except possibly for $z=1$. A notable application is given by the power series
\begin{equation}\label{e:log}
\sum_{n\geq 1} \frac{z^n}{n}\, 
\end{equation}
the power expansion of (a branch of) the [[Logarithmic function|logarithmic function]] $\ln (1-z)$. The Abel criterion implies that the series converges at every $z\neq 1$ with $|z|=1$. Observe that the series reduces to the [[Harmonic series]] at $z=1$, where it diverges. 

====Abel criterion for uniform convergence====
Let $g_n: A \to \mathbb R$ a bounded sequence of functions such that $g_{n+1}\leq g_n$ and $f_n: A\to \mathbb C$ (or more generally $f_n : A \to \mathbb R^k$) a sequence of functions such that $\sum_n f_n$ converges uniformly. Then also the series $\sum_n f_n g_n$ converges uniformly.

====References====
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|valign="top"|{{Ref|Ca}}|| H. Cartan, "Elementary Theory of Analytic Functions of One or Several Complex Variable", Dover (1995). 
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|valign="top"|{{Ref|Fi}}|| G.M. Fichtenholz,    "Differential und Integralrechnung" , '''1''' , Deutsch. Verlag  Wissenschaft.  (1964)
|-
|valign="top"|{{Ref|Ku}}||  L.D. Kudryavtsev,   "Mathematical analysis" , '''1''' , Moscow  (1973)  (In Russian)
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|valign="top"|{{Ref|WW}}||  E.T. Whittaker,   G.N. Watson,   "A course of modern analysis" , '''1–2''' , Cambridge Univ. Press  (1952)
|}