Calculus II Worksheet 8: Convergence and Divergence of Sequences and Series, Assignments of Engineering

Download Calculus II Worksheet 8: Convergence and Divergence of Sequences and Series and more Assignments Engineering in PDF only on Docsity! GEEN 1360 (Calculus II) Worksheet 8 March 8, 2007 Instructions: Be clear in writing up your solutions – show all your work and draw any necessary graphs. Approach these problems step by step, and make sure that everyone in the group can explain how the solution was found. Website: We have a website where you can find these worksheets and their solutions: http://amath.colorado.edu/courses/GEEN1360/2007Spr/ 1. Use ratio and root test to show the convergence of ∞∑ n=1 1 np . What happens? What test do you need to use to show that it converges? 2. For each sequence {an} below, determine if the sequence converges, and if so, to what limit? If the sequence diverges, state this. Be sure to fully support your answer. [Fall ’04] (a) an = sin n n2 (b) an = 3n n3 (c) an = ( n− 2 n )n (d) an = n2/ ln n 3. Determine if the each of the series below converge or diverge. If possible, for each convergent series determine the sum of the series. Be sure to fully support your answer. (a) ∞∑ 0 e−n (b) ∞∑ 3 ln n ln (ln n) (c) ∞∑ 1 2n3n nn (d) ∞∑ 1 9 (3n− 1)(3n + 2) (e) ∞∑ 0 (−1)n 3 4n 4. Does the series defined by the following formula converge or diverge? Be sure to fully support your answer. (a) an = 1/3n if n is odd, an = n/3n if n is even 5. Calculate the infinite sum: 0.4 + 0.16 + 0.064 + .0256 + ... [Fall ’06]