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11Bsn1 - ams/econ 11b supplementary notes ucsc Summation c...

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ams/econ 11b supplementary notes ucsc Summation c 2008, Yonatan Katznelson 1. Notation Many problems in mathematics and its applications (e.g., statistics) involve sums with more than two terms. Writing all of the terms in such a sum can be cumbersome, especially if there are many terms. In some cases we can use ellipses , for example we might write 1 + 2 + 3 + · · · + 100 to indicate the sum of the integers from one to one hundred. But this is a little vague, and in many cases, it might not be clear what terms are missing. The Swiss mathematician Leonard Euler (pronounced oiler ) introduced notation for sums, using the greek letter Σ, which is an upper-case sigma . Definition : Given the terms, A m , A m +1 , A m +2 , . . . , A n , we denote their sum by (1.1) A m + A m +1 + A m +2 + · · · + A n = n X k = m A k . The variable m is called the lower limit of summation , n is called the upper limit of summation and k is called the index of summation . In this notation it is understood that m and n are integers and m n . Furthermore, the index of summation k increases by increments of 1, starting from m and ending at n . The terms { A m , . . . , A n } may be a list of numbers (e.g., data from an experiment), in which case the index k simply enumerates the list. On the other hand, the terms A k may depend on k , i.e., A k may be a function of k — this is the case that we will be considering here. In all of the following examples, the terms of the sum are explicit functions of the index of summation. Examples. 1a. 6 X k =1 k = 1 + 2 + 3 + 4 + 5 + 6. 1b. 100 X j =0 (2 j ) = 0 + 2 + 4 + · · · + 200. Can you tell what terms are missing from the sum 1 + 5 + 11 + · · · + 109? This notation for sums is therefore also called ‘sigma notation’. 1
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2 1c. 15 X i =3 ( i 2 + i - 1) = 11 + 19 + 29 + · · · + 239. Comments: As you can see in the examples, we can use letters other than k to denote the index of summation. The letters i, j and k are common choices, as are m and n (when they’re not being used to denote the limits of summation). Also, as you can see in example 1b. , the lower limit of summation may be zero (or negative). Sigma notation is simply that — notation. It is not a tool for evaluating the sums in question. To evaluate sums, we’ll use the basic properties of addition to develop some simple rules and formulas. On the other hand, the Σ-notation will make these rules and formulas easier to express and understand. 2. Basic rules. The only operation being used in the sum n k = m A k is addition. It follows that all the basic properties of addition hold for such sums. In particular, we can rear- range the terms in a sum, we can collect terms to split a sum into smaller sums and multiplication by a constant factor distributes over a sum. The following three rules illustrate these properties.
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