Can someone explain why there are two summations here?
And why are they exactly the same? I showed a math professor and he
  thinks they're labelled wrong but couldn't figure it out. I don't even
  get why there are two.


When in doubt, check the book's errata (Page 4). 


  $$ \sum_{R(j)} a_j = \Bigg( \lim_{n\rightarrow \infty} \sum_{\color{red}{{\substack{R(j)\\-n\leq j&lt;0}}}} a_j \Bigg ) + \Bigg(\lim_{n\rightarrow \infty} \sum_{{\substack{R(j)\\0\leq j&lt; n}}} a_j \Bigg) $$


Because $R(j)$ is a relation involving $j$, so you must check whether it's satisfied for negative values as well.

The previous paragraph also has a typo:


  $$ \color{red}{\sum_{j=1}^{\infty} a_j} = \sum_{j\geq 1} a_j = a_1 + a_2 + a_3 + \cdots $$