# arithmetic-geometric series

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

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

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

 $1\!+\!q\!+q^{2}\!+\!q^{3}\!+...\,=\frac{1}{1-q},$

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.)

Title arithmetic-geometric series ArithmeticgeometricSeries 2013-03-22 16:02:15 2013-03-22 16:02:15 perucho (2192) perucho (2192) 6 perucho (2192) Derivation msc 40C99