Multiple Choice Questions Chapter 8

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1 Sequences and Series Sultan Qaboos University College of Science Department of Mathematics and Statistics MATH 208 Calculus II Course Coordinator: Ziyad Al-Sharawi (Office 022) Spring 20 Multiple Choice Questions Chapter 8 Remar: The following short practice problems are composed to help Calculus II students grasp the basic ideas of chapter 8. See the acnowledgment and references at the end. Circle the choice below each question that best fits the correct answer:. One of the following sequences is divergent a n = n c n = ( )n n 2. One of the following sequences is not convergent to zero: a n = 3n n b n = 6n n c n = n 2 2n + S n = n ( ) b n = ( ) n 2n3 n 3 + d n = n + 8n 3. One of the following sequences is convergent: a n = ( ) n e n b n = n n, 2,, 8, 6,... c n = ( ) n. One of the following is true: If lim a n = 0, then lim a n 0, If lim f(n) = L, then lim f(x) = L, n x If lim f(x) = L, then lim f(n) = L,. x n If a sequence is bounded, then it must be convergent.. One of the following is true: If a sequence is bounded, then it is convergent. If a sequence is decreasing, then it is convergent. If lim a n = 0, then lim a n = 0. If lim a n exists, then lim a n exists. 6. One of the following is true: If lim n a n = 0, then lim n a n = 0,

2 If lim n a n = 0, then If lim n a n = 0, then If a is divergent. a is convergent. a is divergent, then lim n a What is the definition of a sequence {a n }? A sequence is a set of numbers. A sequence is a relation between two sets. A sequence is a one to one function. A sequence is a function in which the domain is the set of non-negative integers. (E) None of the above. 8. One of the following series is conditionally convergent: ( ) + 2 ( ) ( ) + 9. Which of the following is NOT possible about a sequence a n and its series n= a n? a n converges and a n converges a n converges and a n diverges and a n diverges and n= a n diverges n= a n converges n= a n diverges n= 0. If a sequence a n converges to L 0, then the series conditionally convergent. divergent. absolutely convergent. convergent to L.. If the n-th partial sum S n = n =0 ( ) + a is a = π 2n 3, then lim a equals π 0 does not exist. 2. If a n > 0 and a n+ a n = r < for all values of n, then the series 2 a

3 may converge or diverge. is divergent. is increasing. converges to 3. The series 0. The series a r ( π ) converges to =2 =2 π converges if p π 2 2 π none of these p p > 0 < p p < 0 a n+. If a n > 0 for all n and lim n a n may converge or diverge. is divergent. is convergent by the ratio test. a convergent to r 6. If a is divergent, then 7. If may converge or diverge. is divergent. is conditionally convergent. is convergent a = r <, then the series a a is convergent, then one of the following is true: a is convergent. a is divergent. ( ) a is convergent. the sequence a n is convergent. 8. If 0 < a + a for all and lim a 0, then one of the following is false: a n is decreasing. 3

4 ( ) a is divergent ( ) a is convergent. a is divergent. 9. Suppose that 3 may converge or diverge. is divergent. converges by the integral test. converges to L f(x)dx converges to L 0 and a = f() for all 3. Then 20. One of the following series is absolutely convergent: cos(π) sin(π) 2. One of the following series is convergent: ( e) sin(π) 22. One of the following series is conditionally convergent: ( 2 ( ) 3) cos(π) 23. One of the following series is divergent: ( + 3 ) 3 ( ) ( ) ( + 9)( + 0) + ( + 9) cos(π) 9 a ( ) cos(π) ( ) 2. A student needed to test the series for convergence/divergence. The + sin() student solved the problem as follows: Step : 0 < + sin() for all. Step 2: We now that is a divergent p-series with p =. Step 3: Because is divergent by the comparison test. is divergent, then + sin() What do you thin about the solution of this student?

5 Step is wrong Step 3 is wrong 2. Suppose that a, =0 =0 following statements is true: If If a converges, then =0 b converges, then =0 a and =0 Step 2 is wrong The solution is correct. a n b are series with positive terms and lim = 0. One of the n b n b converges. =0 a converges. =0 b either both converge or both diverge. =0 None of the above. 26. If dx is convergent, then xp is convergent p is convergent. p (E) none of the above. is divergent. p is divergent. p+ 27. If a = 2 + a + 3, then lim equals a 0 none of the above. 28. One of the following is false about power series: A function and its power series agree on the interval of convergence. Every power series has a nonempty interval of convergence. It is possible to find a non-power series that diverges for all values of x. It is possible to find a power series that diverges for all values of x. Acnowledgment: The above questions have been suggested by the following instructors: Aljalila Al-Abri, Ziyad Al-Sharawi, Aref Kamal, Anton Purnama, Medhat Raha, Mohamed Rhouma, Abdulhassan Siddiqi. References:. Smith-Minton, Calculus: Early Transcendental Functions, McGraw Hill, (SQU Special Edition, Boo II). 2. H. Anton, Calculus, Sixth Edition, John Wiley &Sons, 999. Good Luc