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7. What are the values of these sums?

7. What are the values of these sums?

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter2: Equations And Inequalities
Section2.4: Complex Numbers
Problem 37E
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7. What are the values of these sums? 4 a) Σα+1) b) Σ-2)) j=0 k=1 10 8 c) E 3 d) E (2j+1 – 2i) i=1 j=0
Transcribed Image Text:7. What are the values of these sums? 4 a) Σα+1) b) Σ-2)) j=0 k=1 10 8 c) E 3 d) E (2j+1 – 2i) i=1 j=0
Summations Next, we consider the addition of the terms of a sequence. For this we introduce summation notation. We begin by describing the notation used to express the sum of the terms am. dm+l.... dn from the sequence (a,). We use the notation Easisa@j or (read as the sum from j = m to j =n of a ) to represent am + am+1 +...+ an. Here, the variable j is called the index of summation, and the choice of the letter j as the variable is arbitrary: that is, we could have used any other letter, such as i or k. Or, in notation, Here, the index of summation runs through all integers starting with its lower limit m and ending with its upper limit n. A large uppercase Greek letter sigma, E, is used to denote summation. The usual laws for arithmetic apply to summations. For example, when a and b are real numbers, we have E-(ax; + by) = a E- j +bE-Yj, where x, X2, ..., X, and y1. 2..... Ya are real numbers. (We do not present a formal proof of this identity here. Such a proof can be constructed using mathematical induction, a proof method we introduce in Chap- ter 5. The proof also uses the commutative and associative laws for addition and the distributive law of multiplication over addition.) We give some examples of summation notation. EXAMPLE 17 Use summation notation to express the sum of the first 100 terms of the sequence (a,}, where a = 1/j for j = 1, 2, 3. .... Extra Examples Solution: The lower limit for the index of summation is 1, and the upper limit is 100. We write this sum as 100 EXAMPLE 18 What is the value of E-? Solution: We have E-+2+ 3²+ 4² + 5² =I+4+9+ 16+ 25 - 55. EXAMPLE 19 What is the value of -1)? Solution: We have (-1)* = (-1)4 + (-1) +(-1)* +(-1)7 +(-1) =1+(-1) +1+(-1) +1 Sometimes it is useful to shift the index of summation in a sum. This is often done when two sums need to be added but their indices of summation do not match. When shifting an index of summation, it is important to make the appropriate changes in the corresponding summand. This is illustrated by Example 20. EXAMPLE 20 Suppose we have the sum Extra but want the index of summation to run between 0 and 4 rather than from I to 5. To do this, Examples we let k = j -1. Then the new summation index runs from 0 (because k = 1-0=0 when j = 1) to 4 (because k = 5-1= 4 when j= 5), and the term j becomes (k + .Hence, EP =Ek + 1). It is easily checked that both sums are I+4+9+ 16 + 25 - 55. Sums of terms of geometric progressions commonly arise (such sums are called geometric series). Theorem I gives us a formula for the sum of terms of a geometrie progression.
Transcribed Image Text:Summations Next, we consider the addition of the terms of a sequence. For this we introduce summation notation. We begin by describing the notation used to express the sum of the terms am. dm+l.... dn from the sequence (a,). We use the notation Easisa@j or (read as the sum from j = m to j =n of a ) to represent am + am+1 +...+ an. Here, the variable j is called the index of summation, and the choice of the letter j as the variable is arbitrary: that is, we could have used any other letter, such as i or k. Or, in notation, Here, the index of summation runs through all integers starting with its lower limit m and ending with its upper limit n. A large uppercase Greek letter sigma, E, is used to denote summation. The usual laws for arithmetic apply to summations. For example, when a and b are real numbers, we have E-(ax; + by) = a E- j +bE-Yj, where x, X2, ..., X, and y1. 2..... Ya are real numbers. (We do not present a formal proof of this identity here. Such a proof can be constructed using mathematical induction, a proof method we introduce in Chap- ter 5. The proof also uses the commutative and associative laws for addition and the distributive law of multiplication over addition.) We give some examples of summation notation. EXAMPLE 17 Use summation notation to express the sum of the first 100 terms of the sequence (a,}, where a = 1/j for j = 1, 2, 3. .... Extra Examples Solution: The lower limit for the index of summation is 1, and the upper limit is 100. We write this sum as 100 EXAMPLE 18 What is the value of E-? Solution: We have E-+2+ 3²+ 4² + 5² =I+4+9+ 16+ 25 - 55. EXAMPLE 19 What is the value of -1)? Solution: We have (-1)* = (-1)4 + (-1) +(-1)* +(-1)7 +(-1) =1+(-1) +1+(-1) +1 Sometimes it is useful to shift the index of summation in a sum. This is often done when two sums need to be added but their indices of summation do not match. When shifting an index of summation, it is important to make the appropriate changes in the corresponding summand. This is illustrated by Example 20. EXAMPLE 20 Suppose we have the sum Extra but want the index of summation to run between 0 and 4 rather than from I to 5. To do this, Examples we let k = j -1. Then the new summation index runs from 0 (because k = 1-0=0 when j = 1) to 4 (because k = 5-1= 4 when j= 5), and the term j becomes (k + .Hence, EP =Ek + 1). It is easily checked that both sums are I+4+9+ 16 + 25 - 55. Sums of terms of geometric progressions commonly arise (such sums are called geometric series). Theorem I gives us a formula for the sum of terms of a geometrie progression.
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