It's a number of math problems from my calculus 2 class,
Calculus II Math 2414 Exam IV Ch 9 Houston Community College – Katy Campus, Fall 2021 Name : Signature : You have 1hr 45 min to work on your exam. You may use a scientific calculator on this test. All work must be done in pencil. Clear, detailed work must be shown for each problem. If an answer blank is provided, write your answer and only your answer in it. Otherwise box your final answer Important : Problems with insufficient or incorrect supporting work will result in a deduc- tion of points even if the final answer is correct. 1 Problem 1, 10 points, 1 point each 1. True or False If the sequence {sn} is convergent, then lim n→∞ sn = 0. 2. True or False If f(x) is a related function of the sequence {sn} and there is a real number L for which lim n→∞ f(x) = L, then {sn} converges. 3. True or False To use an algebraic ratio to show that the sequence {sn} is increasing, show that sn+1 sn ≥ 0 for all n ≥ 1. 4. True or False A series converges if and only if its sequence of partial sums converge. 5. True or False A geometric series ∞∑ n=1 arn, a 6= 0, converges if |r| ≤ 1. 6. True or False The harmonic series ∞∑ n=1 1 n converges because lim n→∞ = 0. 7. True or False If lim n→∞ an = 0, then the series ∞∑ n=1 an converges. 8. True or False The series ∞∑ n=1 n3 diverges. 9. True or False Let f be a function defined on the interval [1,∞) that is continuous, positive, and decreasing on its domain. Let an = f(k) for all positive integers k. Then the series ∞∑ k=1 ak converges if and only if the improper integral ∫ ∞ 1 f(x) dx converges. 10. True or False If ∞∑ n=1 an and ∞∑ n=1 bn are both series of positive terms and if lim a→∞ an bn = L, where L is a positive real number, then the series to be tested converges. Problem 11, 10 points The first few terms of a sequence are given, find an expression for the nth term of the sequence, assuming the indicated pattern continues for all n. 1 2 ,− 1 3 , 1 4 ,− 1 5 , 1 6 , · · · Problem 12, 10 points Use a related function to show the given sequence converges. Find its limit. { n2 − 4 n2 + n− 2 } 2 Problem 13, 10 points Determine whether the sequence converges or diverges. If it converges, find its limit.{ 3n + 1 4n } Problem 14, 10 points Show that the following sequence diverges : {1 + (−1)n} Problem 15, 10 points Find the sum of the telescoping series: ∞∑ k=1 ( 1 k + 2 − 1 k + 3 ) Problem 16, 10 points Determine whether the sequence converges or diverges. If it converges, find its limit. 1 + 1 4 + 1 16 + · · ·+ ( 1 4 )n + · · · Problem 17, 10 points Express each repeating decimal as a rational number by using a geometric series 4.2855555 · · · (Hint : 4.28555 · · · = 4.28 + 0.00555 · · · ) Problem 18, 10 points Determine whether the series converges or diverges : 1 + 1 2 √ 2 + 1 3 √ 3 + 1 4 √ 4 + · · · Problem 19, 10 points Use the ”Limit Comparison Test” to determine whether the series ∞∑ n=1 3 √ n + 2 √ n3 + 3n2 + 1 3