\begin{definition}[Definition:Arithmetic-Geometric Series]
An '''arithmetic-geometric series''' is a series whose underlying sequence is an arithmetic-geometric sequence:
{{begin-eqn}}
{{eqn | l = S_n
      | r = \sum_{k \mathop = 0}^{n - 1} \paren {a + k d} r^k
      | c = 
}}
{{eqn | r = a + \paren {a + d} r + \paren {a + 2 d} r^2 + \cdots + \paren {a + \paren {n - 1} d}r^{n-1}
      | c = 
}}
{{end-eqn}}
\end{definition}