Sum of Infinite Arithmetic-Geometric Sequence

Theorem
Let $\left \langle{a_k}\right \rangle$ be an arithmetic-geometric progression defined as:
 * $a_k = \left({a + k d}\right) r^k$ for $n = 0, 1, 2, \ldots$

Let:
 * $\left \vert {r}\right \vert < 1$

where $\left \vert {r}\right \vert$ denotes the absolute value of $r$.

Then:
 * $\displaystyle \sum_{n \mathop = 0}^\infty \left({a + k d}\right) r^k = \frac a {1 - r} + \frac {r d} {\left({1 - r}\right)^2}$

Proof
From Sum of Arithmetic-Geometric Progression, we have:
 * $\displaystyle s_n = \sum_{k \mathop = 0}^{n - 1} \left({a + k d}\right) r^k = \frac {a \left({1 - r^n}\right)} {1 - r} + \frac {r d \left({1 - n r^{n - 1} + \left({n - 1}\right) r^n}\right)} {\left({1 - r}\right)^2}$

We have that $\left \vert {r}\right \vert < 1$.

So by Sequence of Powers of Number less than One:
 * $r^n \to 0$ as $n \to \infty$

and
 * $r^{n - 1} \to 0$ as $n - 1 \to \infty$

Hence $s_n \to \dfrac a {1 - r} + \dfrac {r d} {\left({1 - r}\right)^2}$ as $n \to \infty$.

The result follows.