Notation: The Summation and Product Symbols An oversized Greek capital letter sigma is often used to denote a summation, as in Pn i=1 i. This particular expression represents the sum of the integers from 1 to n; that is, it stands for the sum 1 + 2 + 3 + · · · + n. More generally, we can sum any function f(i) of the summation index i. (Of course, the index could be some symbol other than i.) The expression Pb i=a f(i) stands for f(a) + f(a + 1) + f(a + 2) + · · · + f(b)

 

For example, Pm j=2 j 2 stands for the sum 4 + 9 + 16 + · · · + m2 . Here, the function f is “squaring, and we used index j instead of i. As a special case, if b < a, then there are no terms in the sum Pb i=a f(i), and the value of the expression, by convention, is taken to be 0. If b = a, then there is exactly one term, that for i = a. Thus, the value of the sum Pa i=a f(i) is just f(a).

 

The analogous notation for products uses an oversized capital pi. The expression Qb i=a f(i) stands for the product f(a) × f(a + 1) × f(a + 2) × · · · × f(b); if b < a, the product is taken to be 1.