Chapter 5.4

\[ \definecolor{ucrblue}{rgb}{0.0627,0.3843,0.6039} \definecolor{ucrgold}{rgb}{0.9686,0.6941,0.2824} \definecolor{ucrred}{rgb}{0.8941,0,0.1686} \definecolor{ucrgreen}{rgb}{0.4706,0.7451,0.1255} \definecolor{ucraccent}{rgb}{1.0000,0.9569,0.8392} \DeclareMathOperator*{\LO}{O} \DeclareMathOperator*{\Lo}{o} \DeclareMathOperator*{\Recip}{Recip} \DeclareMathOperator*{\abs}{abs} \DeclareMathOperator{\pow}{pow} \]

Notes

• Notes

Linguistic Mapping Exercises

• Linguistic Mapping Exercises PDF

Knowledge Checks

1. Evaluate the following finite series: \(\sum\limits_{k=10}^{50}(3k-2).\)
2. Evaluate the following finite series: \(\sum\limits_{k=1}^{40}\left(\tfrac{1}{k+1}-\tfrac{1}{k+2}\right).\)
3. Evaluate the following finite series: \(\sum\limits_{k=2}^{40}\tfrac{1}{k^2+4k+3}.\)
4. Evaluate the following finite series: \(\sum\limits_{k=3}^{50}\left(\tfrac{1}{2}\right)^k.\)
5. Evaluate the following series: \(\sum\limits_{k=2}^{\infty}\tfrac{1}{k^2+4k+3}.\)
6. Evaluate the following series: \(\sum\limits_{k=3}^{\infty}\left(\tfrac{1}{2}\right)^k.\)
7. Show that the following series diverges: \(\sum\limits_{n=1}^{\infty}\tfrac{n^2+3}{4n^2+n+1}.\)
8. Use the comparison test to determine the convergence or divergence of these series:
   1. \(\sum\limits_{n=1}^{\infty}\tfrac{n+1}{n^3+2n+1}\)
   2. \(\sum\limits_{n=1}^{\infty}\tfrac{n}{\sqrt{4n^2+2}}\)
   3. \(\sum\limits_{n=1}^{\infty}\tfrac{3^n+2n}{10^n+100}\)
9. Use the limit comparison test to determine the convergence or divergence of these series:
   1. \(\sum\limits_{n=1}^{\infty}\tfrac{n+1}{n^3+2n+1}\)
   2. \(\sum\limits_{n=1}^{\infty}\tfrac{n}{\sqrt{4n^2+2}}\)
   3. \(\sum\limits_{n=1}^{\infty}n^2\sin\left(\frac{1}{n^3}\right)\)
   4. \(\sum\limits_{n=1}^{\infty}\tfrac{3^n+2n}{10^n+100}\)
10. Use the alternating series test to determine the convergence of the following series and then determine if they are conditionally convergent or absolutely convergent:
    1. \(\sum\limits_{n=1}^{\infty}\tfrac{(-1)^n}{n^2+1}\)
    2. \(\sum\limits_{n=1}^{\infty}\tfrac{(-1)^n}{e^n}\)
    3. \(\sum\limits_{n=1}^{\infty}\tfrac{1}{\sqrt{3n^2+1}}\)
11. Use the ratio test to determine the convergence of the following series: \(\sum\limits_{n=1}^{\infty}\tfrac{3^n}{n!}.\)
12. Use the root test to determine the convergence of the following series: \(\sum\limits_{n=1}^{\infty}\tfrac{3^n}{2^n}.\)

• Knowledge Checks PDF
• Knowledge Checks Solutions

Remember, do the knowledge checks before checking the solutions.

Practice Problems

• Practice Problems

Return

• Return

© Copyright 2025 by the POC Writing Team: Bryan Carrillo, Yat Sun Poon, and David Weisbart. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the POC Writing Team.