Problem Set 11—Infinite Series, Part 2

Purpose

This problem set reinforces your understanding of power and Taylor series, as well as of some convergence tests not covered in earlier problem sets.

Background

This problem set is based on material in sections 10.5 through 10.8 of our textbook. We discussed this material in classes beginning on April 2.

Activity

Solve each of the following problems:

Problem 1

Section 10.5, exercise 10 (determine whether a series with terms of the form 4ⁿ / (3n)ⁿ converges or diverges).

Problem 2

Section 10.5, exercise 20 (determine whether a series with terms of the form n! / 10ⁿ converges or diverges).

Problem 3

Section 10.6, exercise 54 (how many terms of a series with terms of the form (-1)ⁿ⁺¹ n/(n²+1) need to be evaluated to estimate the series’ limit with an error of at most 0.001).

Problem 4

Section 10.7, exercise 2 (find the interval of convergence and related information about the power series with terms of the form (x+5)ⁿ).

Problem 5

Section 10.8, exercise 12 (find a Maclaurin series for f(x) = xe^x).

Problem 6

Section 10.8, exercise 24 (find a Taylor series about a = 1 for f(x) = 2x³ + x² + 3x - 8).

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.