Chapter 4 Problems

Thirty chapter problems: 15 from the original book and 15 added for more practice.

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1. Original problem

   Use the ratio, root, and comparison test to determine the convergence of the following series.

   

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2. Original problem

   Use the ratio, root, and comparison test to determine the convergence of the following series.

   

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3. Original problem

   Use the ratio, root, and comparison test to determine the convergence of the following series.

   

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4. Original problem

   Use the ratio, root, and comparison test to determine the convergence of the following series.

   

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5. Original problem

   Use the ratio, root, and comparison test to determine the convergence of the following series.

   

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6. Original problem

   Use the ratio, root, and comparison test to determine the convergence of the following series.

   

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7. Original problem

   Use the ratio, root, and comparison test to determine the convergence of the following series.

   

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8. Original problem

   Use the ratio, root, and comparison test to determine the convergence of the following series.

   

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9. Original problem

   Use the ratio, root, and comparison test to determine the convergence of the following series.

   

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10. Original problem

    Use the ratio, root, and comparison test to determine the convergence of the following series.

    

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11. Original problem

    Use the ratio, root, and comparison test to determine the convergence of the following series.

    

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12. Original problem

    Use the ratio, root, and comparison test to determine the convergence of the following series.

    

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13. Original problem

    Use the ratio, root, and comparison test to determine the convergence of the following series.

    

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14. Original problem

    Use the ratio, root, and comparison test to determine the convergence of the following series.

    

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15. Original problem

    Use the ratio, root, and comparison test to determine the convergence of the following series.

    

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16. Added problem

    Use the ratio test to determine whether ${\sum }_{n=1}^{\infty }\frac{n!}{5^n}$ converges.

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17. Added problem

    Use the ratio test to determine whether ${\sum }_{n=1}^{\infty }\frac{5^n}{n!}$ converges.

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18. Added problem

    Determine convergence of ${\sum }_{n=1}^{\infty }(\frac{n+1}{3n+2}{)}^n$ using the root test.

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19. Added problem

    Determine convergence of ${\sum }_{n=1}^{\infty }(1+\frac{1}{n}{)}^{n^2}$ using the root test.

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20. Added problem

    Determine convergence of ${\sum }_{n=1}^{\infty }(\frac{n}{n+1}{)}^{n^2}$ using the root test.

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21. Added problem

    Use direct comparison to determine convergence of ${\sum }_{n=1}^{\infty }\frac{1}{n^2+7}$.

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22. Added problem

    Use limit comparison to determine convergence of ${\sum }_{n=1}^{\infty }\frac{1}{\sqrt{n^2+1}}$.

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23. Added problem

    Use limit comparison to determine convergence of ${\sum }_{n=1}^{\infty }\frac{3n^2+4}{n^4+2}$.

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24. Added problem

    Apply the ratio test to ${\sum }_{n=1}^{\infty }\frac{1}{n^2}$. Explain its conclusion, then determine the actual convergence with another valid test.

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25. Added problem

    Determine convergence of ${\sum }_{n=2}^{\infty }\frac{1}{n+\ln n}$ using direct comparison.

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26. Added problem

    Determine convergence of ${\sum }_{n=1}^{\infty }\frac{(n!{)}^2}{\left(3n\right)!}$ with the ratio test.

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27. Added problem

    Determine convergence of ${\sum }_{n=1}^{\infty }\frac{\left(2n\right)!}{(n!{)}^2{5}^n}$ using the ratio test.

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28. Added problem

    For a real parameter $q\ge 0$, determine exactly when ${\sum }_{n=1}^{\infty }{n}^2{q}^n$ converges. Check the boundary cases.

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29. Added problem

    For real $p$, determine convergence of ${\sum }_{n=1}^{\infty }\frac{1}{n^p(n+1)}$ using limit comparison.

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30. Added problem

    Determine convergence of ${\sum }_{n=1}^{\infty }(\frac{n}{n^3+1}+(\frac{1}{4}{)}^n)$ by giving a useful upper bound.

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