arithmetic-geometric series

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

$G_n(q)={\displaystyle \sum _{k=1}^n}{q}^k={\displaystyle \frac{q}{1-q}}(1-q^n),q\ne 1,$(1)

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

$H_n(q):={\displaystyle \sum _{k=1}^n}k{q}^k,q\ne 1,$(2)

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

$H_n(q)={\displaystyle \sum _{k=1}^n}k{q}^k,q{H}_n(q)={\displaystyle \sum _{k=1}^n}k{q}^{k+1}.$

Subtracting,

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

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

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

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

$H_n(q)={\displaystyle \sum _{k=1}^n}k{q}^k={\displaystyle \frac{q}{{(1-q)}^2}}(1-q^n)-{\displaystyle \frac{n{q}^{n+1}}{1-q}}\cdot$(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, (3) is the partial sum of the convergent series

$H(q)=\underset{n\to \mathrm{\infty }}{lim}{H}_n(q)=\underset{n\to \mathrm{\infty }}{lim}{\displaystyle \sum _{k=1}^n}k{q}^k=\underset{n\to \mathrm{\infty }}{lim}\left[{\displaystyle \frac{q}{{(1-q)}^2}}(1-q^n)-{\displaystyle \frac{n{q}^{n+1}}{1-q}}\right],$

that is,

(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.

$$1+q+q^2+q^3+\mathrm{\dots }=\frac{1}{1-q},$$

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

$$1+2q+3q^2+\mathrm{\dots }=\frac{1}{{(1-q)}^2},$$

$$q+2q^2+3q^3+\mathrm{\dots }=\frac{q}{{(1-q)}^2}$$

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