Abel criterion

2020 Mathematics Subject Classification: Primary: 40A05 [MSN][ZBL]

2020 Mathematics Subject Classification: Primary: 30B30 [MSN][ZBL]

2020 Mathematics Subject Classification: Primary: 40A30 [MSN][ZBL]

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, which is also called Abel's lemma or Abel's transformation and it is a discrete version of 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 $\ln (1-z)$. 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

[Ca]	H. Cartan, "Elementary Theory of Analytic Functions of One or Several Complex Variable", Dover (1995).
[Fi]	G.M. Fichtenholz, "Differential und Integralrechnung" , 1 , Deutsch. Verlag Wissenschaft. (1964)
[Ku]	L.D. Kudryavtsev, "Mathematical analysis" , 1 , Moscow (1973) (In Russian)
[WW]	E.T. Whittaker, G.N. Watson, "A course of modern analysis" , 1–2 , Cambridge Univ. Press (1952)