M408D Quest Homework 2-solutions

# M408D Quest Homework 2-solutions - hernandez(ah29758 –...

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Unformatted text preview: hernandez (ah29758) – M408D Quest Homework 2 – pascaleff – (54550) 1 This print-out should have 14 questions. Multiple-choice 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 zeta-function, ζ , defined by ζ ( 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 ζ ? 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 ζ ( 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 ζ is a p-series with p = x , and so converges for all x > 1 and diverges for all x ≤ 1....
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M408D Quest Homework 2-solutions - hernandez(ah29758 –...

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