Sum of Infinite Arithmetic-Geometric Sequence

Theorem
Let $\sequence {a_k}$ be an arithmetic-geometric sequence defined as:
 * $a_k = \paren {a + k d} r^k$ for $n = 0, 1, 2, \ldots$

Let:
 * $\size r < 1$

where $\size r$ denotes the absolute value of $r$.

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

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

We have that $\size r < 1$.

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:
 * $s_n \to \dfrac a {1 - r} + \dfrac {r d} {\paren {1 - r}^2}$

as $n \to \infty$.

The result follows.