arithmetic-geometric series

It is well known that a finite geometric seriesMathworldPlanetmath is given by

Gn(q)=k=1nqk=q1-q(1-qn),q1, (1)

where in general q=reiθ is complex. When we are dealing with such sums it is common to consider the expression

Hn(q):=k=1nkqk,q1, (2)

which we shall call an arithmetic-geometric series. Let us derive a formula for Hn(q).




We will proceed to eliminate the right-hand side sums.


By using (1) and solving for Hn(q), we obtain

Hn(q)=k=1nkqk=q(1-q)2(1-qn)-nqn+11-q (3)

The formula (3) holds in any commutative ring with 1, as long as (1-q) is invertiblePlanetmathPlanetmath. If q is a complex numberMathworldPlanetmathPlanetmath and |q|<1, (3) is the partial sum of the convergent seriesMathworldPlanetmathPlanetmath


that is,

H(q)=k=1kqk=q(1-q)2,|q|<1. (4)

This last result giving the sum of a converging arithmetic-geometric series may be, naturally, obtained also from the sum formula of the converging geometric series, i.e.


when one differentiates both sides with respect to q and then multiplies them by q:


(A power series can be differentiated termwise on the open interval of convergence.)

Title arithmetic-geometric series
Canonical name ArithmeticgeometricSeries
Date of creation 2013-03-22 16:02:15
Last modified on 2013-03-22 16:02:15
Owner perucho (2192)
Last modified by perucho (2192)
Numerical id 6
Author perucho (2192)
Entry type Derivation
Classification msc 40C99