Chapter 5 Problems

Thirty chapter problems: 6 from the original book and 24 added for more practice.

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1. Original problem

   What is required for a series to pass the alternating series test?

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2. Original problem

   What is required for a series to pass the test for absolute convergence?

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3. Original problem

   Do the following sums define alternating series?

   1. 
   2. 

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4. Original problem

   Determine if the following sums converge.

   1. 
   2. 

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5. Original problem

   Determine whether the following series are absolutely convergent, conditionally convergent, or neither.

   1. 
   2. 
   3. 
   4. 

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6. Original problem

   4. Show that the alternating series may be arranged so that it may converge to a value other than the value of example 5.3.2

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7. Added problem

   Classify ${\sum }_{n=1}^{\infty }(-1{)}^{n-1}/\mathrm{sqrt}(n)$ as absolutely convergent, conditionally convergent, or divergent.

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8. Added problem

   Classify ${\sum }_{n=1}^{\infty }(-1{)}^n/n^2$.

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9. Added problem

   Does ${\sum }_{n=1}^{\infty }(-1{)}^{n}n/(n+1)$ converge? Explain.

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10. Added problem

    Classify ${\sum }_{n=2}^{\infty }(-1{)}^n\ln n/n$.

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11. Added problem

    Classify ${\sum }_{n=1}^{\infty }(-1{)}^n/(2n+1)$.

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12. Added problem

    How many terms of ${\sum }_{n=1}^{\infty }(-1{)}^{n-1}/n$ guarantee an error less than 0.01 using the alternating error bound?

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13. Added problem

    Approximate ${\sum }_{n=1}^{\infty }(-1{)}^{n-1}/n$ using four terms and bound the error.

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14. Added problem

    Use the alternating error bound to find the smallest N guaranteeing error less than 0.0001 for ${\sum }_{n=1}^{\infty }(-1{)}^{n-1}/n^2$.

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15. Added problem

    Does every alternating series converge? Give a counterexample and explain.

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16. Added problem

    Find the sum and degree of convergence of ${\sum }_{n=0}^{\infty }(-3/4{)}^n$.

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17. Added problem

    Classify ${\sum }_{n=1}^{\infty }\cos n/n^2$.

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18. Added problem

    Classify ${\sum }_{n=1}^{\infty }(-1{)}^{n}n/2^n$.

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19. Added problem

    Classify ${\sum }_{n=0}^{\infty }(-1{)}^n{3}^n/n!$ as absolutely convergent, conditionally convergent, or divergent.

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20. Added problem

    What does absolute convergence tell you about rearranging a series? Explain using its tail.

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21. Added problem

    Does convergence imply absolute convergence? Give an example.

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22. Added problem

    Classify ${\sum }_{n=2}^{\infty }(-1{)}^n/(n\ln n)$.

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23. Added problem

    Classify ${\sum }_{n=2}^{\infty }(-1{)}^n/(n\left(\ln n{)}^2\right)$.

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24. Added problem

    Can a conditionally convergent series be rearranged to have sum zero? Explain.

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25. Added problem

    Describe how to rearrange the alternating harmonic series to converge to a chosen real number L.

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26. Added problem

    Find the sum of $1-1/2-1/4+1/3-1/6-1/8+1/5-1/10-1/12+\dots$, using one positive term followed by two negatives.

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27. Added problem

    Can rearranging ${\sum }_{n=1}^{\infty }(-1{)}^n/n^2$ change its sum? Why?

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28. Added problem

    Classify ${\sum }_{n=1}^{\infty }(-1{)}^{n-1}n/(n^2+1)$.

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29. Added problem

    For $1-1/2+1/4-1/8+\dots$, compare the actual error after three terms with the alternating error bound.

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30. Added problem

    Find both ${\sum }_{n=0}^{\infty }(-1/3{)}^n$ and ${\sum }_{n=0}^{\infty }\left|\right(-1/3{)}^n|$.

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