Abel’s multiplication rule for series
Cauchy has originally presented the multiplication rule
for two series. His assumption 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
Proof. We consider the corresponding power series
When , they give the series
which we assume to converge. Thus the power series are absolutely convergent for , whence they obey the multiplication rule due to Cauchy:
On the other hand, the sums of the power series (2) are, as is well known, continuous functions on the interval ; the same concerns the right hand side of (3), because for it becomes the third series which we assume convergent. When , 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 , also the limits of both of (3), as , 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|
|Date of creation||2014-11-22 21:19:35|
|Last modified on||2014-11-22 21:19:35|
|Last modified by||pahio (2872)|
|Synonym||Abel’s multiplication rule|