4.3: Products
4.3.1: Product Notation
Read Section 1 on pages 1 - 11. ∏ is the symbol for product. If p1 , p2, ... , pk ... is a sequence, the series p1 , p1 p2 , p1 p2 p3 , p1 p2 , ... , pk is written ∏ p1 p2 ... pk. If you look over the exercises for section 1, you will see that this reading applies to ∏ examples as well as ∑ examples. This reading also applies to topics in Subunit 4.3.2.
4.3.2: Computing Products
Massachusetts Institute of Technology: Srini Devadas and Eric Lehman's "Sums and Approximations" URL
Study Section 3.4 "Approximating 1 + x."
Product computation is similar to the evaluation of the product of numbers in arithmetic. Product or multiplication is a binary operation that is commutative [i.e. a times b = a b = b a]; associative [i.e. (a b) c = a (b c)]; and has an identity (i.e. a times 1 = 1). In addition, there are some tricks that can be used.
Some simple properties of products are:
1. a ∏ pi = ∏ a pi, where the ∏ ranges over a set of positive integers i.
2. ∏ pi ∏ qij = ∏ pi qj, where the product symbols varies over ranges for i and j.
Read Theorem 11 and example 17 on pages IS-27 through IS-30. In doing calculation involving series, we work with sums (since a series is the sum of the succeeding terms of a sequence). In working these calculations, we also encounter the product of terms, for example when determining whether a series is bounded.