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arithmetic-geometric series

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

$\displaystyle G_n(q)=\sum_{k=1}^nq^k=\frac{q}{1-q}(1-q^n), \qquad q\neq 1,$

(1)

where in general $q=re^{i\theta}$ is complex. When we are dealing with such sums it is common to consider the expression

$\displaystyle H_n(q):=\sum_{k=1}^n kq^k, \qquad q\neq 1,$

(2)

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

$\displaystyle H_n(q)=\sum_{k=1}^n kq^k, \qquad qH_n(q)=\sum_{k=1}^n kq^{k+1}.$

Subtracting,

$\displaystyle (1-q)H_n(q)=\sum_{k=1}^n kq^k-\sum_{k=1}^n kq^{k+1}= \sum_{k=1}^n kq^k-\sum_{k=2}^{n+1}(k-1)q^k= \sum_{k=1}^n kq^k-\sum_{k=2}^n(k-1)q^k-nq^{n+1}.$

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

$\displaystyle (1-q)H_n(q)=q+\sum_{k=2}^n q^k -nq^{n+1}= \sum_{k=1}^n q^k-nq^{n+1}.$

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

$\displaystyle H_n(q)=\sum_{k=1}^n kq^k=\frac{q}{(1-q)^2}(1-q^n)-\frac{nq^{n+1}}{1-q}\:\cdot$

(3)

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

$\displaystyle H(q)=\lim_{n\to\infty}H_n(q)=\lim_{n\to\infty}\sum_{k=1}^n kq^k= \lim_{n\to\infty}\bigg[\frac{q}{(1-q)^2}(1-q^n)-\frac{nq^{n+1}}{1-q}\bigg],$

that is,

$\displaystyle H(q)=\sum_{k=1}^\infty kq^k=\frac{q}{(1-q)^2},\, \qquad \vert q\vert<1.$

(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. $$1\!+\!q\!+q^2\!+\!q^3\!+...\, = \frac{1}{1-q},$$ when one differentiates both sides with respect to $q$ and then multiplies them by $q$ : $$1\!+\!2q\!+\!3q^2\!+...\, = \frac{1}{(1\!-\!q)^2},$$ $$q\!+\!2q^2\!+\!3q^3\!+...\, = \frac{q}{(1\!-\!q)^2}$$ (A power series can be differentiated termwise on the open interval of convergence.)

arithmetic-geometric series is owned by Pedro A. Fernandez, Thomas Foregger, J. Pahikkala.

How to Cite This Entry

Pedro A. Fernandez, Thomas Foregger, J. Pahikkala. "arithmetic-geometric series" (version 3). PlanetMath.org. Freely available at http://planetmath.org/ArithmeticGeometricSeries.html.

Classification

AMS MSC:

40C99 (Sequences, series, summability :: General summability methods :: Miscellaneous)

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