The sum notation

Consider an expression like

We will meet similar expressions quite often in the book, and a nice shorthand notation for this expression is

which should be read as the sum of “x subscript i,” for i ranging from 1 to 4. Sometimes, the sum limits can be symbolic, as in

We may even consider an infinite sum like

In this case, we should wonder whether the expression makes sense, since summing an infinite number of terms may result in a sum going to infinity. A thorough study of the involved issues requires the theory of mathematical series; in Section 2.12 we deal with a few examples that are most relevant in applications. In some cases, we might wish to skip values corresponding to some subscript:

Finally, when variables have two subscripts, we may consider double sums like

In this example we also see that the order of the two sums is irrelevant and they may be swapped:

We will always take for granted that this is the case, even though some care should be taken with infinite sums. When sum limits are irrelevant, we may use streamlined notations like

The latter notation comes in handy when we want to exclude terms of the form x₁₁, x₂₂, x₃₃, etc.