Problem Set 9—Infinite Sequences and Series

Purpose

This problem set reinforces your understanding of the basic features of infinite sequences and series.

Background

This problem set is based on material in sections 10.1 and 10.2 of our textbook. We discussed this material in class during the week of March 24.

Activity

Solve each of the following problems:

Problem 1

Theorem 4 in section 10.1, says, roughly, that if a_n = f(n) and lim_x→∞f(x) = L, then lim_n→∞a_n = L. Is an analogous statement true if lim_x→∞f(x) = ±∞? Justify your answer.

Problem 2

Section 10.1, exercise 6 (list the first 4 terms of the sequence defined by a_n = ( 2ⁿ-1) / 2ⁿ).

Problem 3

Section 10.1, exercise 28 (determine whether the sequence defined by a_n = ( n + (-1)ⁿ ) / n converges or diverges). Confirm your answer by using a calculator or mathematical programming language to plot the first 50 terms of the sequence; if you believe that the sequence converges, include in the same plot a flat line whose y value is the supposed limit.

Problem 4

Section 10.1, exercise 54 (determine whether the sequence defined by a_n = ( 1 - (1/n) )ⁿ converges or diverges). Confirm your answer by using a calculator or mathematical programming language to plot the first 50 terms of the sequence; if you believe that the sequence converges, include in the same plot a flat line whose y value is the supposed limit.

Problem 5

Section 10.2, exercise 8 (write out the first 8 terms of the infinite series whose terms have the form 1/4ⁿ, n ≥ 2; determine whether the series converges or diverges, and, if it converges, find the sum). Confirm your answer by using a calculator or mathematical programming language to plot the first 50 partial sums of the series; if you believe that the series converges, include in the same plot a flat line whose y value is the supposed sum.

Follow-Up

I will grade this exercise in a face-to-face meeting with you. During this meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Please bring a written solution to the exercise to your meeting, as reading through it will help me know what to focus on in the rest of the meeting.

Sign up for a meeting via Google calendar. If you worked in a group on this exercise, the whole group should schedule a single meeting with me. Please make the meeting 15 minutes long, and schedule it to finish before the end of the “Grade By” date above.

My basic expectation in grading this exercise is that your solution will show that you understand how to solve each problem, although there may be arithmetic or copying mistakes, inefficient solution methods, incorrect or irrelevant statements incidental to the solution, or similar minor mistakes. If you understand how to solve all the problems and have no minor errors, I will consider the solution to be in between “what I expect” and “surprisingly beyond expectations.” I will consider solutions to be 3/4, 1/2, 1/4, or none of what I expect according to what (rough) fraction of the problems your solution shows understanding of, although I will raise grades slightly if it is clear by the end of your grading meeting that you have come to understand things you didn’t understand when you arrived at the meeting.