Abel’s multiplication rule for series

Cauchy has originally presented the multiplicationPlanetmathPlanetmath rule

j=1ajk=1bk=n=1(a1bn+a2bn-1++anb1) (1)

for two series.  His assumptionPlanetmathPlanetmath was that both of the multiplicand series should be absolutely convergent.  Mertens (1875) lightened the assumption requiring that both multiplicands should be convergent but at least one of them absolutely convergent (see the parent (http://planetmath.org/MultiplicationOfSeries) entry).  N. H. Abel’s most general form of the multiplication rule is the

TheoremMathworldPlanetmath.  The rule (1) for multiplication of series with real or complex terms is valid as soon as all three of its series are convergent.

Proof.  We consider the corresponding power series

j=1ajxj,k=1bkxk. (2)

When  x=1,  they give the series


which we assume to convergePlanetmathPlanetmath.  Thus the power series are absolutely convergent for |x|<1, whence they obey the multiplication rule due to Cauchy:

j=1ajxjk=1bkxk=n=1(a1bn+a2bn-1++anb1)xn+1. (3)

On the other hand, the sums of the power series (2) are, as is well known, continuous functionsMathworldPlanetmathPlanetmath on the interval[0, 1];  the same concerns the right hand side of (3), because for  x=1  it becomes the third series which we assume convergent.  When  x1-, we infer that


and that the limit of the right hand side of (3) is the right hand side of (1).  Since the equation (3) is true for |x|<1,  also the limits of both of (3), as  x1-, are equal.  Therefore the equation (1) is in with the assumptions of the theorem.


  • 1 E. Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset III. Toinen osa.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1940).
Title Abel’s multiplication rule for series
Canonical name AbelsMultiplicationRuleForSeries
Date of creation 2014-11-22 21:19:35
Last modified on 2014-11-22 21:19:35
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Theorem
Classification msc 40A05
Synonym Abel’s multiplication rule
Related topic AbelsLimitTheorem
Related topic NielsHenrikAbel