(BQ) Part 2 book "Calculus early transcendental functions" has contents: Conics, parametric equations, and polar coordinates; vectors and the geometry of space; vector valued functions; functions of several variables; multiple integration; vector analysis. | 680 Chapter 9 Infinite Series 8. Using the Alternating Series Test function Ά 1, f ͑x͒ ϭ sin x , x The graph of the 13. Deriving Identities Derive each identity using the appropriate geometric series. (a) 1 ϭ . . . (b) xϭ0 1 ϭ . . . x > 0 is shown below. Use the Alternating Series Test to show that the improper integral ͵ ϱ 1 f ͑x͒ dx converges. y 1 π 2π 3π 14. Population Consider an idealized population with the characteristic that each member of the population produces one offspring at the end of every time period. Each member has a life span of three time periods and the population begins with 10 newborn members. The following table shows the population during the first five time periods. x 4π Time Period Age Bracket 1 2 3 4 5 0–1 10 10 20 40 70 10 10 20 40 10 10 20 40 70 130 −1 9. Conditional and Absolute Convergence For what values of the positive constants a and b does the following series converge absolutely? For what values does it converge conditionally? b a b a b a b . . . ϩ Ϫ ϩ Ϫ ϩ Ϫ ϩ 2 3 4 5 6 7 8 2–3 Total 10 20 The sequence for the total population has the property that Sn ϭ SnϪ1 ϩ SnϪ2 ϩ SnϪ3, n > 3. Find the total population during each of the next five time periods. 10. Proof (a) Consider the following sequence of numbers defined recursively. a1 ϭ 3 a2 ϭ Ί3 a3 ϭ Ί3 ϩ Ί3 15. Spheres Imagine you are stacking an infinite number of spheres of decreasing radii on top of each other, as shown in the figure. The radii of the spheres are 1 meter, 1͞Ί2 meter, 1͞Ί3 meter, and so on. The spheres are made of a material that weighs 1 newton per cubic meter. (a) How high is this infinite stack of spheres? Ӈ (b) What is the total surface area of all the spheres in the stack? anϩ1 ϭ Ί3 ϩ an Write the decimal approximations for the first six terms of this sequence. Prove that the sequence converges, and find its limit. (c) Show that the weight of the .
