# Question on the Range in Limit Notation of an Infinite Sum

I'm studying Knuth's The Art of Computer Programming, Volume 1 (3rd ed.) and came across the following on page 27:

The precise meaning of $\sum_{R(j)} a_j$ is therefore

$$\sum_{R(j)} a_j = (\lim_{n\to\infty} \sum_{R(j), -n\leq j < 0} a_j) + (\lim_{n\to\infty} \sum_{R(j), 0\leq j < n} a_j)$$

On the second limit, why does the range exclude n? I see there's a symmetry to the two inequalities when written this way, but don't see why -n would be included in the range while n is not. It calls to mind the ranges of two's complement binary integers, but I wouldn't think that has any relevance here.

I've looked through the errata for the book on Knuth's web site, and see the equation mentioned in a correction, but only to flip the order of the terms on the right-hand-side. So, having already had some amount of attention, I don't believe there is a typo in the equation.

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## 1 Answer

It simply doesn't make any difference. Presumably Knuth wants both ranges to be of length $n$, but you can replace $<$ with $\leq$ and vice versa without effecting the limit.

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+1. The only thing is to avoid counting $a_0$ twice. – Did Sep 10 '11 at 20:13