Convergence Tests

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 11.7 – Strategy for Testing Series

Practice 04

Determine if the following series are convergent or divergent. Explain.

1. ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}(-1{)}^n(0.3{)}^n}$$\displaystyle \sum_{n=0}^{\infty}{(-1)^n(0.3)^n}$

2. ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}{n}^{-1/4}}$$\displaystyle \sum_{n=1}^{\infty}{n^{-1/4}}$

3. ${\displaystyle \sum _{n=2}^{\mathrm{\infty }}\frac{n}{(n^3+2{)}^{3/7}}}$$\displaystyle \sum_{n=2}^{\infty}{\frac{n}{(n^3+2)^{3/7}}}$

4. ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}\frac{n!}{e^n}}$$\displaystyle \sum_{n=0}^{\infty}{\frac{n!}{e^n}}$

5. ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}\frac{n^2}{n+1}}$$\displaystyle \sum_{n=0}^{\infty}{\frac{n^2}{n+1}}$

6. ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{\left(\frac{3}{\pi }\right)}^n}$$\displaystyle \sum_{n=0}^{\infty}{\left(\frac{3}{\pi}\right)^n}$

7. ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}\frac{n}{e^n}}$$\displaystyle \sum_{n=0}^{\infty}{\frac{n}{e^n}}$

8. ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{3}{n^{0.9999}}}$$\displaystyle \sum_{n=1}^{\infty}{\frac{3}{n^{0.9999}}}$

9. ${\displaystyle \sum _{n=3}^{\mathrm{\infty }}\frac{n^2-\sqrt{n}}{4-n^2}}$$\displaystyle \sum_{n=3}^{\infty}{\frac{n^2-\sqrt{n}}{4-n^2}}$

10. ${\displaystyle \sum _{n=2}^{\mathrm{\infty }}\frac{1}{n\sqrt{n}-n}}$$\displaystyle \sum_{n=2}^{\infty}{\frac{1}{n\sqrt{n}-n}}$

11. ${\displaystyle \sum _{n=2}^{\mathrm{\infty }}\frac{1}{(\ln (n){)}^n}}$$\displaystyle \sum_{n=2}^{\infty}{\frac{1}{(\ln{(n)})^n}}$

12. ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n+1}\left(\frac{3^n}{2^{n+2}}\right)}$$\displaystyle \sum_{n=1}^{\infty}{(-1)^{n+1}\left(\frac{3^n}{2^{n+2}}\right)}$

13. ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}{\left(\frac{5n+2}{7n+2}\right)}^n}$$\displaystyle \sum_{n=1}^{\infty}{\left(\frac{5n+2}{7n+2}\right)^n}$

14. ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}(-1{)}^n\left(\frac{n}{n^2+2}\right)}$$\displaystyle \sum_{n=1}^{\infty}{(-1)^{n}\left(\frac{n}{{n^2+2}}\right)}$

15. ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{n^3+1}{3^n}}$$\displaystyle \sum_{n=1}^{\infty}{\frac{n^3+1}{3^n}}$

16. ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{\sqrt[3]{n+5}}{\sqrt{n+2}}}$$\displaystyle \sum_{n=1}^{\infty}{\frac{\sqrt[3]{n+5}}{\sqrt{n+2}}}$

17. ${\displaystyle \sum _{n=2}^{\mathrm{\infty }}\frac{1}{n\ln (n)}}$$\displaystyle \sum_{n=2}^{\infty}{\frac{1}{n\ln{(n)}}}$

18. ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}\frac{5^n{n}^3}{n!}}$$\displaystyle \sum_{n=0}^{\infty}{\frac{5^nn^3}{n!}}$

19. ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{n}^2{e}^{-n}}$$\displaystyle \sum_{n=0}^{\infty}{n^2e^{-n}}$

20. ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{2n}}{n^n}}$$\displaystyle \sum_{n=1}^{\infty}{\frac{(-1)^{2n}}{n^n}}$

Use Technology to Verify Convergence or Divergence of a Series

NOTE: On all Assessments, you may use technology to verify your series convergence/divergence. You must show all Calculus II related work.

Mathematica

1
(* Example from Practice 04-12 *)
2
SumConvergence[(-1)^(n+1) 3^n/2^(n+2), n]

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.

2. SumConvergence[ ] has at least two arguments:

   1. The $n^{\text{th}}$$n^{\text{th}}$-term of the series;

   2. $n$$n$ where $n$$n$ is the index variable of the series.

3. For help on using the SumConvergence[ ] function:

   1. In Mathematica, execute the code: $\text{?SumConvergence}$$\text{?SumConvergence}$

   2. Click on $\vee$$\or$ near the bottom-left of output

   3. Click on local

   4. Read how to use the SumConvergence[ ] function – you will be able to copy-paste code.

4. Remember, correct Mathematica code will be all black except for variables.

5. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

6. Mathematica will return “True” (series converges) or “False” (series diverges).

Homework

At this time, you should be able to complete the following assignments:

• Section 11.7: # 1, 3, 5, 7, 9, 11, 13, 14, 18, 23, 25.

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Created: Tuesday, 17 November 2020 7:25 EDT Last Modified: Tuesday, 30 May 2023 – 23:08 (EDT)