This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: hernandez (ah29758) M408D Quest Homework 2 pascaleff (54550) 1 This printout should have 14 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. 001 10.0 points Compare the values of the series A = summationdisplay n = 1 4 n 2 . 4 and the improper integral B = integraldisplay 1 4 x 2 . 4 dx . 1. A < B 2. A = B 3. A > B correct Explanation: In the figure 1 2 3 4 5 . . . a 1 a 2 a 3 a 4 the bold line is the graph of the function f ( x ) = 4 x 2 . 4 on [1 , ) and the area of each of the rectan gles is one of the values of a n = 4 n 2 . 4 . Clearly from this figure we see that a 1 = f (1) > integraldisplay 2 1 f ( x ) dx, a 2 = f (2) > integraldisplay 3 2 f ( x ) dx , while a 3 = f (3) > integraldisplay 4 3 f ( x ) dx, a 4 = f (4) > integraldisplay 5 4 f ( x ) dx , and so on for all n . Consequently, A = summationdisplay n =1 4 n 2 . 4 > integraldisplay 1 4 x 2 . 4 dx = B . 002 10.0 points Which of the following series are convergent: A . summationdisplay n = 1 2 n 2 / 3 B . summationdisplay n = 1 3 n + 1 C . 1 + 1 4 + 1 9 + 1 16 + . . . 1. none of them 2. B and C only 3. A only 4. B only 5. A and C only 6. A and B only 7. C only correct hernandez (ah29758) M408D Quest Homework 2 pascaleff (54550) 2 8. all of them Explanation: By the Integral test, if f ( x ) is a positive, decreasing function, then the infinite series summationdisplay n =1 f ( n ) converges if and only if the improper integral integraldisplay 1 f ( x ) dx converges. Thus for the three given series we have to use an appropriate choice of f . A. Use f ( x ) = 2 x 2 / 3 . Then integraldisplay 1 f ( x ) dx is divergent. B. Use f ( x ) = 3 x + 1 . Then integraldisplay 1 f ( x ) dx is divergent (log integral). C. Use f ( x ) = 1 x 2 . Then integraldisplay 1 f ( x ) dx is convergent. keywords: convergent, Integral test, 003 10.0 points The Riemann zetafunction, , defined by ( x ) = summationdisplay k =1 1 k x is very important in number theory where it is used to study the distribution of prime numbers. What is the natural domain of ? 1. natural domain = ( , 1] 2. natural domain = ( , 1) 3. natural domain = (1 , ) correct 4. natural domain = (0 , 1] 5. natural domain = [1 , ) Explanation: The natural domain of a function f is the set of ALL values of x for which f ( x ) is well defined. In the case of the the Riemann zeta function ( x ) = summationdisplay k =1 1 k x , therefore, its natural domain is the set of all values of x for which the series converges. But is a pseries with p = x , and so converges for all x > 1 and diverges for all x 1....
View Full
Document
 Spring '07
 TextbookAnswers

Click to edit the document details