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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.

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    $\begingroup$ Don't use images as main content of your post. This makes your question impossible to search and inaccessible to the visually impaired; we don't like that. Please transcribe text and mathematics (note that you can use LaTeX) and don't forget to give proper attribution to your sources! $\endgroup$ – D.W. May 2 '17 at 7:44
  • $\begingroup$ Please don't use the 'answer' box to try to clarify your question. Instead, click the 'edit' link under your question to revise it to what it should have been from the start. It looks like you might have created multiple accounts. I encourage you to merge them: cs.stackexchange.com/help/merging-accounts. This will help ensure you can edit you question and post comments. $\endgroup$ – D.W. May 2 '17 at 7:46
  • $\begingroup$ Finally, this appears to be off-topic here, as it is a pure math question. Generally speaking, pure math questions belong on Mathematics, unless there is some reason that they are best answered from a computer science perspective, or there is a strong connection to computer science; in that case, I recommend that you describe the connection in the question. $\endgroup$ – D.W. May 2 '17 at 7:48
  • $\begingroup$ Can you be more specific about what you mean by "wouldn't let me"? What exactly was the error? $\endgroup$ – D.W. May 2 '17 at 7:49
  • $\begingroup$ Big important part of the text is kidding to infer anything $\endgroup$ – Eugene May 2 '17 at 9:11
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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).

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

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:

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

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    $\begingroup$ Shouldn't the modification of the first formula be without the $a_0$? (and with $0 \leq j <n$ below the second summation sign)? $\endgroup$ – user53923 May 2 '17 at 15:18
  • $\begingroup$ @user53923 You are correct! Edited the typos. $\endgroup$ – Aristu May 2 '17 at 15:28

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