jmc

A $\frac{1}{p}$ -array is a structured, infinite, collection of numbers. For example, a $\frac{1}{3}$ -array is constructed as follows: $$\begin{array}{rl}1\frac{1}{3}\frac{1}{9}\frac{1}{27}& \cdots \\ \frac{1}{6}\frac{1}{18}\frac{1}{54}& \cdots \\ \frac{1}{36}\frac{1}{108}& \cdots \\ \frac{1}{216}& \cdots \\ & \ddots \end{array}$$ In general, the first entry of each row is $\frac{1}{2p}$ times the first entry of the previous row. Then, each succeeding term in a row is $\frac{1}{p}$ times the previous term in the same row. If the sum of all the terms in a $\frac{1}{2008}$ -array can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find the remainder when $m+n$ is divided by $2008$.