Problem Set 10—Infinite Series

Purpose

This problem set reinforces your basic understanding of infinite series and of some of their tests for convergence.

Background

This problem set is based on material in sections 10.2 through 10.4 of our textbook. We discussed section 10.2 in class on March 26, and will discuss the convergence tests in sections 10.3 and 10.4 on March 31 and April 1.

Activity

Solve each of the following problems:

Problem 1

Section 10.2, exercise 52 (determine whether the infinite series whose terms are of the form (-1)ⁿ⁺¹n converges or diverges, and find the sum if it converges).

Problem 2

Section 10.2, exercise 54 (determine whether the infinite series whose terms are of the form cosnπ / 5ⁿ converges or diverges, and find the sum if it converges).

Problem 3

Section 10.2, exercise 82 (no matter what real number you are given, is it possible to define an infinite series with that number as its limit).

Problem 4

Section 10.3, exercise 2 (determine whether the infinite series whose terms are of the form 1 / n^0.2 converges or diverges). In addition, if you believe the series converges, use a calculator or mathematical programming language to plot the first 50 partial sums of the series and estimate what value it converges to.

Problem 5

Section 10.3, exercise 12 (determine whether the infinite series whose terms are of the form e^-n converges or diverges). In addition, if you believe the series converges, use a calculator or mathematical programming language to plot the first 50 partial sums of the series and estimate what value it converges to.

Problem 6

Section 10.4, exercise 2 (determine whether the infinite series whose terms are of the form (n-1) / (n⁴+2) converges or diverges). In addition, if you believe the series converges, use a calculator or mathematical programming language to plot the first 50 partial sums of the series and estimate what value it converges to.

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.