Difference between revisions of "Abel criterion"
From Encyclopedia of Mathematics
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| − | + | {{MSC|40A05}} | |
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| + | {{TEX|done}} | ||
| − | + | 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]]. | ||
| − | is convergent. | + | ====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 | + | ====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. | |
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====References==== | ====References==== | ||
| − | + | {| | |
| + | |- | ||
| + | |valign="top"|{{Ref|Ca}}|| H. Cartan, "Elementary Theory of Analytic Functions of One or Several Complex Variable", Dover (1995). | ||
| + | |- | ||
| + | |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) | ||
| + | |- | ||
| + | |valign="top"|{{Ref|WW}}|| E.T. Whittaker, G.N. Watson, "A course of modern analysis" , '''1–2''' , Cambridge Univ. Press (1952) | ||
| + | |} | ||
Latest revision as of 12:44, 10 December 2013
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) |
How to Cite This Entry:
Abel criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abel_criterion&oldid=30926
Abel criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abel_criterion&oldid=30926
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article