Can someone explain why there are two summations here? The following quote is from the book "The art of computer programming":


  (..) sentence  would presumably be used only if either $j$ or $k$ (not both) has exterior significance.
  
  In most cases we will use notation (2) only when the sum is finite - that is, when only a finite number of values $j$ satisfy $R(j)$ and have $a_j \neq 0$. If an infinite sum is required, for example
  $$\sum_{j=1}^{\infty} = \sum_{j \geq 1}a_j = a_1 + a_2 + a_3+\cdots$$
  with infinitely many nonzero terms, the techniques of calculus must be employed; the precise meaning of (2) is then
  $$\qquad\qquad \sum_{R(j)} a_j = \left(\lim_{n\rightarrow\infty} \sum_{R(j) \atop 0\leq j \leq n} a_j\right) + \left(\lim_{n\rightarrow\infty} \sum_{R(j) \atop 0\leq j \leq n} a_j\right),\qquad\qquad(3)$$
  
  (...)


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. 

Sorry if the question isn't clear enough. I'm referring to the two infinite sums. As far as (2) is concerned he is only referring to what is on the left hand side of the equation with the two infinite sums. Its just a way of representing any possible series. So essentially my question is how does making any series go to infinity make two of them? Or am I misunderstanding?

Also I tried to post this in math stack exchange and it wouldn't let me so I came here since its from the book, the art of computer programming.
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:


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