Calculus II Worksheet 7 - Convergence and Series, Assignments of Engineering

Download Calculus II Worksheet 7 - Convergence and Series and more Assignments Engineering in PDF only on Docsity! GEEN 1360 (Calculus II) Worksheet 7 February 29, 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. Determine whether the sequences below converge or diverge. If it converges, find the limit. (a) an = sin n n (b) an = sinh (lnn) (c) an = ∫ n 1 1 xp dx, p > 1 (d) an = 1 n ∫ n 1 1 x dx 2. If x1 = cos (0) sin (1) and xn+1 = max{xn, cos (n) sin (n + 1)}, does the sequence converge or diverge? 3. A ball is dropped from a height of ten feet and bounces. Each bounce is 34 of the height of the bounce before. Thus after the ball hits the floor for the first time, the ball rises to a height of 10( 34 ) = 7.5 feet, and after it hits the floor for the second time, it rises to a height of 7.5( 34 ) = 10( 3 4 ) 2 = 5.625 feet. (a) Find an expression for the height to which the ball rises after it hits the floor for the nth time. (b) Find an expression for the total vertical distance the ball has traveled when it hits the floor for the first, second, third, and fourth times. (c) Find an expression for the total vertical distance the ball has traveled. 4. Do the following sequences {an}∞1 converge or diverge? If the sequence converges, find the value. Be sure to fully explain your reasoning. [Spring ’05] (a) an = cos ( nπ 2 ) (b) an = cn, where c > 0 is a constant (c) an = nb ecn , where b > 0 and c > 0 are constants (d) an = ( 1 + b n )n , where b is a constant 5. Although the harmonic series does not converge, the partial sums grow very, very slowly. Take a right-hand sum approximating the integral of f(x) = 1/x on the interval [1,n], with ∆x = 1, to show that 1 2 + 1 3 + 1 4 + ... + 1 n < ln n If a computer could add a million terms of the harmonic series each second, estimate the sum after one year.