What is the proper notation for a pattern?

Question

Which of the following notations is proper, and which is not?

$$x = A_1 + \cdots + A_n$$ $$y = A_1 + A_2 + \cdots + A_n$$ $$z = A_1 + A_2 + \cdots + A_{n-1} + A_n$$

I have seen all of these styles used in various places, but some of these would yield the wrong result when the number of elements in $A$ is small. For example, if $A=\{ 1, 2 \}$, $y$ and $z$ would not have the value I intended, which is $3$.

Where can I read more about this notation, so I know the right number of things to show in a pattern, and what to call these things?

Answer 1

In Concrete Mathematics by D.E. Knuth, R.L. Graham and O. Patashnik the chapter 2: Sums, section 2.1: Notation starts with:

In Chapter 1 we encountered the sum of the first $n$ integers, which we wrote out as $$1+2+3+\cdots+(n-1)+n$$ The ` $\cdots$ ' in such formulas tells us to complete the pattern established by the surrounding terms. Of course we have to watch out for sums like $1 + 7 + \cdots + 41.7$, which are meaningless without a mitigating context.

On the other hand, the inclusion of terms like $3$ and $(n - 1)$ was a bit of overkill; the pattern would presumably have been clear if we had written simply $1 + 2 + \cdots + n$. Sometimes we might even be so bold as to write just $1 + \cdots + n$.

We'll be working with sums of the general form \begin{align*}a_1 + a_2 + \cdots + a_n \end{align*} where each $a_k$ is a number that has been defined somehow. This notation has the advantage that we can see the whole sum, almost as if it were written out in full, if we have a good enough imagination.

The sigma notation $\sum_{k=1}^n a_k$ and its many different shapes is thoroughly explained after that.

Answer 2

The right notation, in this case, is sigma notation: $$z = \sum_{i=1}^n A_i.$$ It's rigorously defined, and completely unambiguous.

That said, patterns are still used because sometimes they can make an expression more readable, even if they are not 100% clear. Certain operations like changing the order of summation (particularly when the bounds of the sum are not "rectangular") tend to be easier to follow when written out in one of the styles above (especially when the reader is less experienced).

So, none of them are "right" or "proper", but all three are acceptable, provided that nobody can reasonably misread the pattern.