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Mathematics LibreTexts

8.E: Applications of Sequences and Series (Exercises)

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    8.1: Sequences

    Terms and Concepts

    1. Use your own words to define a sequence. 

    2. The domain of a sequence is the _____ numbers.

    3. Use your own words to describe the range of a sequence.

    4. Describe what it means for a sequence to be bounded.

    Problems

    In Exercises 5-8, give the first five terms of the given sequence.

    5. \({a_n}=\left \{ \frac{4^n}{(n+1)!}\right \}\)

    6. \({b_n}=\left \{ \left ( -\frac{3}{2}\right )^n\right \}\)

    7. \({c_n}=\left \{ -\frac{n^{n+1}}{n+2}\right \}\)

    8. \({d_n}=\left \{ \frac{1}{\sqrt{5}}\left ( \left ( \frac{1+\sqrt{5}}{2}\right )^n - \left ( \frac{1-\sqrt{5}}{2}\right )^2\right )\right \}\)

    In Exercises 9-12, determine the \(n^{th}\) term of the given sequence.

    9. 4, 7, 10, 13, 16, ...

    10. \(3,\,-\frac{3}{2},\,\frac{3}{4},\,-\frac{3}{8},...\)

    11. 10, 20, 40, 80, 160, ...

    12. \(1,\, 1,\, \frac{1}{2},\,\frac{1}{6},\, \frac{1}{24},\,\frac{1}{120},...\)

    In Exercises 13-16, use the following information to determine the limit of the given sequences.

    • \({a_n} = \left \{ \frac{2^n-20}{2^n}\right \};\quad \lim\limits_{n\to \infty}a_n=1\)
    • \({b_n} = \left \{ \left ( 1+\frac{2}{n}\right )^n \right \};\quad \lim\limits_{n\to \infty}b_n=e^2\)
    • \({c_n} = \left \{ \frac{2^n-20}{2^n}\right \};\quad \lim\limits_{n\to \infty}c_n=0\)

    13. \({a_n} = \left \{ \frac{2^n-20}{7\cdot 2^n}\right \}\)

    14. \({a_n}={3b_n-a_n}\)

    15. \({a_n}=\left \{ \sum\limits (3/n) \left ( 1=\frac{2}{n}\right )^n\right \}\)

    16. \({a_n}=\left \{  \left ( 1=\frac{2}{n}\right )^{2n}\right \}\)

    In Exercises 17-28, determine whether the sequence converges or diverges. If convergent, give the limit of the sequence.

    17. \({a_n}=\left \{  (-1)^n \frac{n}{n+1}\right \}\)

    18. \({a_n}=\left \{   \frac{4n^2-n+5}{3n^2+1}\right \}\)

    19. \({a_n}=\left \{   \frac{4^n}{5^n}\right \}\)

    20. \({a_n}=\left \{  \frac{n-1}{n}-\frac{n}{n-1}\right \},\,n\ge 2\)

    21. \({a_n}=\left \{  \ln (n)\right \}\)

    22. \({a_n}=\left \{   \frac{3n}{\sqrt{n^2+1}}\right \}\)

    23. \({a_n}=\left \{  \left ( 1+\frac{1}{n}\right )^n\right \}\)

    24. \({a_n}=\left \{  5-\frac{1}{n}\right \}\)

    25. \({a_n}=\left \{   \frac{(-1)^{n+1}}{n}\right \}\)

    26. \({a_n}=\left \{   \frac{1.1^n}{n}\right \}\)

    27. \({a_n}=\left \{   \frac{2n}{n+1}\right \}\)

    28. \({a_n}=\left \{  (-1)^n \frac{n^2}{2^n-1}\right \}\)

    In Exercises 29-34, determine whether the sequence is bounded, bounded above, bounded below, or none of the above.

    29. \({a_n}={\sum\limits n}\)

    30. \({a_n}={\tan n}\)

    31. \({a_n}={(-1)^n \frac{3n-1}{n}}\)

    32. \({a_n}=\left \{ \frac{3n^2-1}{n}\right \}\)

    33. \({a_n}={n \cos n}\)

    34. \({a_n}={2^n-n!}\)

    In Exercises 35-38, determine whether the sequence is monotonically increasing or decreasing. If it is not, determine if there is an m such that it is monotonic for all \(n \ge m\).

    35. \({a_n}=\left \{ \frac{n}{n+2}\right \}\) 

    36. \({a_n}=\left \{ \frac{n^2-6n+9}{n}\right \}\) 

    37. \({a_n}=\left \{ (-1)^n\frac{1}{n^3}\right \}\) 

    38. \({a_n}=\left \{ \frac{n^2}{2^n}\right \}\) 

    39. Prove Theorem 56; that is, use the definition of the limit of a sequence to show that if \(\lim\limits_{n\to \infty}|a_n|=0\), then \(\lim\limits_{n\to\infty}a_n=0\).

    40. Let \({a_n}\text{ and }{b_n}\) be sequences such that \(\lim\limits_{n\to\infty}a_n=L\text{ and }\lim\limits_{n\to\infty}b_n=K\).
    (a) Show that if \(a_n<b_n\) for all n, then \(L\le K\).
    (b) Give an example where \(L=K\).

    41. Prove the Squeeze Theorem for sequences: Let \({a_n}\text{ and }{b_n}\) be such that \(\lim\limits_{n\to\infty}a_n =L\text{ and }\lim\limits_{n\to\infty}b_n =L\), and let \({c_n}\) be such that \(a_n\le c_n \le b_n\) for all n. Then \(\lim\limits_{n\to\infty}c_n=L\)

    8.2: Infinite Series

    Terms and Concepts

    1. Use your own words to describe how sequences and series are related.

    2. Use your own words to define a partial sum.

    3. Given a series \(\sum\limits_{n=1}^{\infty}a_n\)m describe the two sequences related to the series that are important.

    4. Use your own words to explain what a geometric series is.

    5. T/F: If \({a_n}\) is convergent, then \(\sum\limits_{n=1}^{\infty}a_n\) is also convergent.

    Problems

    In Exercises 6-13, a series \(\sum\limits_{n=1}^{\infty}a_n\) is given.
    (a) Give the first 5 partial sums of the series.
    (b) Give a graph of the first 5 terms of
    \(a_n\text{ and }S_n\) on the same axes.

    6. \(\sum\limits_{n=1}^{\infty}\frac{(-1)^n}{n}\)

    7. \(\sum\limits_{n=1}^{\infty}\frac{1}{n^2}\)

    8. \(\sum\limits_{n=1}^{\infty}\cos (\pi n)\)

    9. \(\sum\limits_{n=1}^{\infty}n\)

    10. \(\sum\limits_{n=1}^{\infty}\frac{1}{n!}\)

    11. \(\sum\limits_{n=1}^{\infty}\frac{1}{3^n}\)

    12. \(\sum\limits_{n=1}^{\infty}\left ( -\frac{9}{10}\right )^n\)

    13. \(\sum\limits_{n=1}^{\infty}\left ( \frac{1}{10}\right )^n\)

    In Exercises 14-19, use Theorem 63 to show the given series diverges.

    14. \(\sum\limits_{n=1}^{\infty}\frac{3n^2}{n(n+2)}\)

    15. \(\sum\limits_{n=1}^{\infty}\frac{2^n}{n^2}\)

    16. \(\sum\limits_{n=1}^{\infty}\frac{n!}{10^n}\)

    17. \(\sum\limits_{n=1}^{\infty}\frac{5^n-n^5}{5^n+n^5}\)

    18. \(\sum\limits_{n=1}^{\infty}\frac{2^n+1}{2^{n+1}}\)

    19. \(\sum\limits_{n=1}^{\infty}\left ( 1+\frac{1}{n}\right )^n\)

    In Exercises 20-29, state whether the given series converges or diverges.

    20. \(\sum\limits_{n=1}^{\infty}\frac{1}{n^5}\)

    21. \(\sum\limits_{n=0}^{\infty}\frac{1}{5^n}\)

    22. \(\sum\limits_{n=0}^{\infty}\frac{6^n}{5^n}\)

    23. \(\sum\limits_{n=1}^{\infty}n^{-4}\)

    24. \(\sum\limits_{n=1}^{\infty}\sqrt{n}\)

    25. \(\sum\limits_{n=1}^{\infty}\frac{10}{n!}\)

    26. \(\sum\limits_{n=1}^{\infty}\left ( \frac{1}{n!}+\frac{1}{n}\right )\)

    27. \(\sum\limits_{n=1}^{\infty}\frac{2}{(2x+8)^2}\)

    28. \(\sum\limits_{n=1}^{\infty}\frac{1}{2n}\)

    29. \(\sum\limits_{n=1}^{\infty}\frac{1}{2n-1}\)

    In Exercises 30-44, a series is given.
    (a) Find a formula for
    \(S_n\), the \(n^{th}\) partial sum of the series.
    (b) Determine whether the series converges or diverges. If it converges, state what it converges to.

    30. \(\sum\limits_{n=0}^{\infty}\frac{1}{4^n}\)

    31. \(1^3+2^3+3^3+4^3+...\)

    32. \(\sum\limits_{n=1}^{\infty}(-1)^n n\)\)

    33. \(\sum\limits_{n=0}^{\infty}\frac{5}{2^n}\)

    34. \(\sum\limits_{n=1}^{\infty}e^{-n}\)

    35. \(1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\frac{1}{81}+...\)

    36. \(\sum\limits_{n=1}^{\infty}\frac{1}{n(n+1)}\)

    37. \(\sum\limits_{n=1}^{\infty}\frac{3}{n(n+2)}\)

    38. \(\sum\limits_{n=1}^{\infty}\frac{1}{(2x-1)(2x+1)}\)

    39. \(\sum\limits_{n=1}^{\infty}\ln \left (\frac{n}{n+1}\right )\)

    40. \(\sum\limits_{n=1}^{\infty}\frac{2n+1}{n^2(n+1)^2}\)

    41. \(\frac{1}{1\cdot 4}+\frac{1}{2\cdot 5}+\frac{1}{3\cdot 6}+\frac{1}{4\cdot 7}+...\)

    42. \(2 +\left ( \frac{1}{2}+\frac{1}{3}\right ) +\left ( \frac{1}{4}+\frac{1}{9}\right )+\left ( \frac{1}{8}+\frac{1}{27}+...\right )\)

    43. \(\sum\limits_{n=2}^{\infty}\frac{1}{n^2-1}\)

    44. \(\sum\limits_{n=0}^{\infty}\left ( \sin 1 \right )^n\)

    45. Break the Harmonic Series into the sum of the odd and even terms:
    \(\sum\limits_{n=1}^{\infty}\frac{1}{n} = \sum\limits_{n=1}^{\infty}\frac{1}{2n-1}+\sum\limits_{n=1}^{\infty}\frac{1}{2n}\).
    The goal is to show that each of the series on the right diverge.
    (a) Show why \(\sum\limits_{n=1}^{\infty}\frac{1}{2n-1}>\sum\limits_{n=1}^{\infty}\frac{1}{2n}\). (Compare each \(n^{th}\) partial sum.) 
    (b) Show why \(\sum\limits_{n=1}^{\infty}\frac{1}{2n-1}<1+\sum\limits_{n=1}^{\infty}\frac{1}{2n}\)
    (c) Explain why (a) and (b) demonstrate that the series of odd terms is convergent, if, and only if, the series of even terms is also convergent. (That is, show both converge or both diverge.)
    (d) Explain why knowing the Harmonic Series is divergent determines that the even and odd series are also divergent.

    46. Show the series \(\sum\limits_{n=1}^{\infty} \frac{n}{(2n-1)(2n+1)}\) diverges.

    8.3: Integral and Comparison Tests

    Terms and Concepts

    1. In order to apply the Integral Test to a sequence \({A_n}\), the function \(a(n)=a_n\) must be _____, _____ and  _____.

    2.T/F: The Integral Test can be used to determine the sum of a convergent series.

    3.What test(s) in this section do not work well with factorials?

    4. Suppose \(\sum\limits_{n=0}^{\infty} a_n\) is convergent, and there are sequences \({b_n}\) and \({c_n}\) such that \(b_n \le a_n \le c_n\) for all n. What can be said about the series \(\sum\limits_{n=0}^{\infty}b_n \text{ and }\sum\limits_{n=0}^{\infty}c_n\)?

    Problems

    In Exercises 5-12, use the Integral Test to determine the convergence of the given series.

    5. \(\sum\limits_{n=1}^{\infty}\frac{1}{2^n}\)

    6. \(\sum\limits_{n=1}^{\infty}\frac{1}{n^4}\)

    7. \(\sum\limits_{n=1}^{\infty}\frac{n}{n^2+1}\)

    8. \(\sum\limits_{n=2}^{\infty}\frac{1}{n\ln n}\)

    9. \(\sum\limits_{n=1}^{\infty}\frac{1}{n^2+1}\)

    10. \(\sum\limits_{n=2}^{\infty}\frac{1}{n(\ln n)^2}\)

    11. \(\sum\limits_{n=1}^{\infty}\frac{n}{2^n}\)

    12. \(\sum\limits_{n=1}^{\infty}\frac{\ln n}{n^3}\)

    In Exercises 13-22, use the Direct Comparison Test to determine the convergence of the given series; state what series is used for comparison.

    13. \(\sum\limits_{n=1}^{\infty}\frac{1}{n^2+3n-5}\)

    14. \(\sum\limits_{n=1}^{\infty}\frac{1}{4^n+n^2-n}\)

    15. \(\sum\limits_{n=1}^{\infty}\frac{\ln n}{n}\)

    16. \(\sum\limits_{n=1}^{\infty}\frac{1}{n!+n}\)

    17. \(\sum\limits_{n=2}^{\infty}\frac{1}{\sqrt{n^2-1}}\)

    18. \(\sum\limits_{n=1}^{\infty}\frac{1}{\sqrt{n}-2}\)

    19. \(\sum\limits_{n=1}^{\infty}\frac{n^2+n+1}{2^n}\)

    20. \(\sum\limits_{n=1}^{\infty}\frac{2^n}{5^n+10}\)

    21. \(\sum\limits_{n=2}^{\infty}\frac{n}{n^2-1}\)

    22. \(\sum\limits_{n=2}^{\infty}\frac{1}{n^2\ln n}\)

    In Exercises 23-32, use the Limit Comparison Test to determine the convergence of the given series; state what series is used for comparison.

    23. \(\sum\limits_{n=1}^{\infty}\frac{1}{n^2-3n+5}\)

    24. \(\sum\limits_{n=1}^{\infty}\frac{1}{4^n-n^2}\)

    25. \(\sum\limits_{n=4}^{\infty}\frac{\ln n}{n-3}\)

    26. \(\sum\limits_{n=1}^{\infty}\frac{1}{\sqrt{n^2+n}}\)

    27. \(\sum\limits_{n=1}^{\infty}\frac{1}{n+\sqrt{n}}\)

    28. \(\sum\limits_{n=1}^{\infty}\frac{n-10}{n^2+10n+10}\)

    29. \(\sum\limits_{n=1}^{\infty}\sin \left ( 1/n \right )\)

    30. \(\sum\limits_{n=1}^{\infty}\frac{n+5}{n^3-5}\)

    31. \(\sum\limits_{n=1}^{\infty}\frac{\sqrt{n}+3}{n^2+17}\)

    32. \(\sum\limits_{n=1}^{\infty}\frac{1}{\sqrt{n}+100}\)

    In Exercises 33-40, determine the convergence of the given series. State the test used; more than one test may be appropriate.

    33. \(\sum\limits_{n=1}^{\infty}\frac{n^2}{2^n}\)

    34. \(\sum\limits_{n=1}^{\infty}\frac{1}{(2n+5)^3}\)

    35. \(\sum\limits_{n=1}^{\infty}\frac{n!}{10^n}\)

    36. \(\sum\limits_{n=1}^{\infty}\frac{\ln n}{n!}\)

    37. \(\sum\limits_{n=1}^{\infty}\frac{1}{3^n+n}\)

    38. \(\sum\limits_{n=1}^{\infty}\frac{n-2}{10n+5}\)

    39. \(\sum\limits_{n=1}^{\infty}\frac{3^n}{n^3}\)

    40. \(\sum\limits_{n=1}^{\infty}\frac{\cos \left ( 1/n\right )}{\sqrt{n}}\)

    41. Given that \(\sum\limits_{n=1}^{\infty}a_n\) converges, state which of the following series converges, may converge, or does not converge.
    (a) \(\sum\limits_{n=1}^{\infty}\frac{a_n}{n}\)
    (b) \(\sum\limits_{n=1}^{\infty}a_na_{n+1}\)
    (c) \(\sum\limits_{n=1}^{\infty}\left ( a_n \right )^2\)
    (d) \(\sum\limits_{n=1}^{\infty}na_n\)
    (e) \(\sum\limits_{n=1}^{\infty}\frac{1}{a_n}\)

    8.4: Ratio and Root Tests

    Terms and Concepts

    1. The Ratio Test is not effective when the terms of a sequence only contain ______ functions

    2. The Ratio Test is most effective when the terms of a sequence contains ____ and/or _____ functions.

    3. What three convergence tests do not work well with terms containing factorials?

    4. The Root Test works particularly well on series where each term is _____ to a _____.

    Problems

    In Exercises 5-14, determine the convergence of the given series using the Ratio Test. If the Ratio Test is inconclusive, state so and determine the convergence with another test.

    5. \(\sum\limits_{n=0}^{\infty}\frac{2n}{n!}\)

    6. \(\sum\limits_{n=0}^{\infty}\frac{5^n-3n}{4^n}\)

    7. \(\sum\limits_{n=0}^{\infty}\frac{n!10^n}{(2n)!}\)

    8. \(\sum\limits_{n=1}^{\infty}\frac{5^n+n^4}{7^n+n^2}\)

    9. \(\sum\limits_{n=1}^{\infty}\frac{1}{n}\)

    10. \(\sum\limits_{n=1}^{\infty}\frac{1}{3n^3+7}\)

    11. \(\sum\limits_{n=1}^{\infty}\frac{10\cdot 5^n}{7^n-3}\)

    12. \(\sum\limits_{n=1}^{\infty}n\cdot \left (\frac{3}{5}\right )^n\)

    13. \(\sum\limits_{n=1}^{\infty}\frac{2\cdot 4\cdot 6\cdot 8 \cdot\cdot\cdot 2n}{3\cdot 6\cdot 9 \cdot 12 \cdot\cdot\cdot 3n}\)

    14. \(\sum\limits_{n=1}^{\infty}\frac{n!}{5\cdot 10\cdot 15 \cdot\cdot\cdot (5n)}\)

    In Exercises 15-24, determine the convergence of the given series using the Root Test. If the Root Test is inconclusive, state so and determine convergence with another test.

    15. \(\sum\limits_{n=1}^{\infty}\left (\frac{2n+5}{3n+11}\right )^n\) 

    16. \(\sum\limits_{n=1}^{\infty}\left (\frac{0.9n^2-n-3}{n^2+n+3}\right )^n\)

    17. \(\sum\limits_{n=1}^{\infty}\frac{2^nn^2}{3^n}\)

    18. \(\sum\limits_{n=1}^{\infty}\frac{1}{n^n}\)

    19. \(\sum\limits_{n=1}^{\infty}\frac{3^n}{n^22^{n+1}}\)

    20. \(\sum\limits_{n=1}^{\infty}\frac{4^{n+7}}{7^n}\)

    21. \(\sum\limits_{n=1}^{\infty}\left (\frac{n^2-n}{n^2+n}\right )\)

    22. \(\sum\limits_{n=1}^{\infty}\left ( \frac{1}{n}-\frac{1}{n^2}\right )^2\)

    23. \(\sum\limits_{n=1}^{\infty}\frac{1}{\left ( \ln n\right )^2}\)

    24. \(\sum\limits_{n=1}^{\infty}\frac{n^2}{\left ( \ln n\right )^n}\)

    In Exercises 25-34, determine the convergence of the given series. State the test used; more than one test may be appropriate.

    25. \(\sum\limits_{n=1}^{\infty}\frac{n^2+4n-2}{n^3+4n^2-3n+7}\)

    26. \(\sum\limits_{n=1}^{\infty}\frac{n^44^n}{n!}\)

    27. \(\sum\limits_{n=1}^{\infty}\frac{n^2}{3^n+n}\)

    28. \(\sum\limits_{n=1}^{\infty}\frac{3^n}{n^n}\)

    29. \(\sum\limits_{n=1}^{\infty}\frac{n}{\sqrt{n^2+4n+1}}\)

    30. \(\sum\limits_{n=1}^{\infty}\frac{n!n!n!}{(3n)!}\)

    31. \(\sum\limits_{n=1}^{\infty}\frac{1}{\ln n}\)

    32. \(\sum\limits_{n=1}^{\infty}\left ( \frac{n+2}{n+1}\right )\)

    33. \(\sum\limits_{n=1}^{\infty}\frac{n^3}{\left ( \ln n\right )^n}\)

    34. \(\sum\limits_{n=1}^{\infty}\left (\frac{1}{n}-\frac{1}{n+2}\right )\)

    8.5: Alternating Series and Absolute Convergence

    Terms and Concepts

    1. Why is \(\sum\limits_{n=1}^{\infty}\sin n\) not an alternating series?

    2. A series \(\sum\limits_{n=1}^{\infty} (-1)^n a_n\) converges when \({a_n}\) is _____ _____ and \(\lim\limits_{n\to \infty}a_n=\)_____. 

    3. Give an example of a series when \(\sum\limits_{n=0}^{\infty}a_n\) converges but \(\sum\limits_{n=0}^{\infty}|a_n|\) does not.

    4. The sum of a _____ convergent series can be changed by rearranging the order of its terms.

    Problems

    In Exercises 5-20, an alternating series \(\sum\limits_{n=i}^{\infty}a_n\) is given.
    (a) Determine if the series converges or diverges.
    (b) Determine if
    \(\sum\limits_{n=0}^{\infty}|a_n|\) converges or diverges.
    (c) If
    \(\sum\limits_{n=0}^{\infty}a_n\) converges, determine if the convergence is conditional or absolute.

    5. \(\sum\limits_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^2}\)

    6. \(\sum\limits_{n=1}^{\infty}\frac{(-1)^{n+1}}{\sqrt{n!}}\)

    7. \(\sum\limits_{n=0}^{\infty}(-1)^n\frac{n+5}{3n-5}\)

    8. \(\sum\limits_{n=1}^{\infty}(-1)^n\frac{2n}{n^2}\)

    9. \(\sum\limits_{n=0}^{\infty}(-1)^{n+1}\frac{3n+5}{n^2-3n+1}\)

    10. \(\sum\limits_{n=1}^{\infty}\frac{(-1)^n}{\ln n+1}\)

    11. \(\sum\limits_{n=2}^{\infty}(-1)^n\frac{n}{\ln n}\)

    12. \(\sum\limits_{n=1}^{\infty}\frac{(-1)^{n+1}}{1+3+5+...+ (2n-1)}\)

    13. \(\sum\limits_{n=1}^{\infty}\cos (\pi n)\)

    14. \(\sum\limits_{n=1}^{\infty}\frac{\sin ((n+1/2)\pi)}{n\ln n}\)

    15. \(\sum\limits_{n=0}^{\infty}\left ( -\frac{2}{3}\right )^n\)

    16. \(\sum\limits_{n=0}^{\infty}(-e)^{-n}\)

    17. \(\sum\limits_{n=0}^{\infty}\frac{(-1)^nn^2}{n!}\)

    18. \(\sum\limits_{n=0}^{\infty}(-1)^n2^{-n^2}\)

    19. \(\sum\limits_{n=0}^{\infty}\frac{(-1)^n}{\sqrt{n}}\)

    20. \(\sum\limits_{n=1}^{\infty}\frac{(-1000)^2}{n!}\)

    Let \(S_n\) be the n\(^{th}\) partial sum of a series. In Exercises 21-24, a convergent alternating series is given and a value of n. Compute \(S_n\text{ and }S_{n+1}\) and use these values to find bounds on the sum of the series.

    21. \(\sum\limits_{n=1}^{\infty}\frac{(-1)^n}{\ln  (n+1)},\quad n=5\)

    22. \(\sum\limits_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^4},\quad n=4\)

    23. \(\sum\limits_{n=0}^{\infty}\frac{(-1)^n}{n!},\quad n=6\)

    24. \(\sum\limits_{n=0}^{\infty}\left ( -\frac{1}{2}\right )^n,\quad n=9\)

    In Exercises 25-28, a convergent alternating series is given along with its sum and a value of \(\epsilon\). Use Theorem 71 to find n such that the n\(^{th}\) partial sum of the series is within \(\epsilon\) of a sum of the series.

    25. \(\sum\limits_{n=0}^{\infty}\frac{(-1)^{n+1}}{n^4}=\frac{7\pi^4}{720},\quad \epsilon =0.001\)

    26. \(\sum\limits_{n=0}^{\infty}\frac{(-1)^n}{n!}=\frac{1}{e},\quad \epsilon =0.0001\)

    27. \(\sum\limits_{n=0}^{\infty}\frac{(-1)^n}{2n+1}=\frac{\pi}{4},\quad \epsilon =0.001\)

    28. \(\sum\limits_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}=\cos 1,\quad \epsilon =10^{-8}\)

    8.6: Power Series

    Terms and Concepts

    1. We adopt the convection that \(x^0 =\)_____, regardless of the value of x.

    2. What is the difference between the radius of convergence and the interval of convergence?

    3. If the radius of convergence of \(\sum\limits_{n=0}^{\infty}a_xx^n\) is 5, what is the radius of convergence of \(\sum\limits_{n=1}^{\infty}n\cdot a_nx^{n-1}\)?

    4. If the radius of convergence of \(\sum\limits_{n=0}^{\infty}a_nx^n\) is 5, what is the radius of convergence of \(\sum\limits_{n=0}^{\infty}(=1)^na_nx^n\)?

    Problems

    In Exercises 5-8, write out the sum of the first 5 terms of the given power series.

    5. \(\sum\limits_{n=0}^{\infty}2^nx^n\)

    6. \(\sum\limits_{n=1}^{\infty}\frac{1}{n^2}x^n\)

    7. \(\sum\limits_{n=0}^{\infty}\frac{1}{n!}x^n\)

    8. \(\sum\limits_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}x^{2n}\)

    In Exercises 9-24, a power series is given.
    (a) Find the radius of convergence.
    (b) Find the interval of convergence.

    9. \(\sum\limits_{n=0}^{\infty}\frac{(-1)^{n+1}}{n!}x^n\)

    10. \(\sum\limits_{n=0}^{\infty}nx^n\)

    11. \(\sum\limits_{n=1}^{\infty}\frac{(-1)^n(x-3)^n}{n}\)

    12. \(\sum\limits_{n=0}^{\infty}\frac{(x+4)^n}{n!}\)

    13. \(\sum\limits_{n=0}^{\infty}\frac{x^n}{2^n}\)

    14. \(\sum\limits_{n=0}^{\infty}\frac{(-1)^n(x-5)^n}{10^n}\)

    15. \(\sum\limits_{n=0}^{\infty}5^n (x-1)^n\)

    16. \(\sum\limits_{n=0}^{\infty}(-2)^nx^n\)

    17. \(\sum\limits_{n=0}^{\infty}\sqrt{n}x^n\)

    18. \(\sum\limits_{n=0}^{\infty}\frac{n}{3^n}x^n\)

    19. \(\sum\limits_{n=0}^{\infty}\frac{3^n}{n!}(x-5)^n\)

    20. \(\sum\limits_{n=0}^{\infty}(-1)^n n! (x-10)^n\)

    21. \(\sum\limits_{n=1}^{\infty}\frac{x^n}{n^2}\)

    22. \(\sum\limits_{n=1}^{\infty}\frac{(x+2)^n}{n^3}\)

    23. \(\sum\limits_{n=0}^{\infty}n! \left ( \frac{x}{10}\right )^n\)

    24. \(\sum\limits_{n=0}^{\infty}n^2 \left (\frac{x+4}{4}\right )^n\)

    In Exercises 25-30, a function \(f(x) = \sum\limits_{n=0}^{\infty}a_nx^n\) is given.
    (a) Give a power series for
    \(f'(x)\) and its interval of convergence.
    (b) Give a power series for
    \(f;(x)\,dx\) and its interval of convergence.

    25. \(\sum\limits_{n=0}^{\infty}nx^n\)

    26. \(\sum\limits_{n=1}^{\infty}\frac{x^n}{n}\)

    27. \(\sum\limits_{n=0}^{\infty}\left ( \frac{x}{2}\right )^n\)

    28. \(\sum\limits_{n=0}^{\infty}(-3x)^n\)

    29. \(\sum\limits_{n=0}^{\infty}\frac{(-1)^nx^{2n}}{(2n)!}\)

    30. \(\sum\limits_{n=0}^{\infty}\frac{(-1)^nx^n}{n!}\)

    In Exercises 31-36, give the first 5 terms of the series that is a solution to the given differential equation.

    31. \(y' =3y,\quad y(0)=1\)

    32. \(y' =5y,\quad y(0)=5\)

    33. \(y' =y^2,\quad y(0)=1\)

    34. \(y' =y+1,\quad y(0)=1\)

    35. \(y'' =-y,\quad y(0)=0,\,y'(0)=1\)

    36. \(y'' =2y,\quad y(0)=1,\,y'(0)=1\)

    8.7: Taylor Polynomials

    Terms and Concepts

    1. What is the difference between a Taylor polynomial and a Maclaurin polynomial?

    2. T/F: In general, \(p_n(x)\) approximates \(f(x)\) better and better as n gets larger.

    3. For some function \(f(x)\), the Maclaurin polynomial of degree 3 is \(p_4(x)=6+3x-4x^2+5x^3-7x^4\). What is \(p_2(x)\)?

    4. For some function \(f(x)\), the Maclaurin polynomial of degree 3 is \(p_4(x)=6+3x-4x^2+5x^3-7x^4\). What is \(f'''(0)\)?

    Problems

    In Exercises 5-12, find the Maclaurin polynomial of degree n for the given function.

    5. \(f(x) = e^{-x},\quad n=3\).

    6. \(f(x) = \sin x,\quad n=8\).

    7. \(f(x) = x\cdot e^{x},\quad n=5\).

    8. \(f(x) = \tan x,\quad n=6\).

    9. \(f(x) = e^{2x},\quad n=4\).

    10. \(f(x) = \frac{1}{1-x},\quad n=4\).

    11. \(f(x) = \frac{1}{1+x},\quad n=4\).

    12. \(f(x) = \frac{1}{1-x},\quad n=7\).

    In Exercises 13-20, find the Taylor polynomial of degree n, at \(x=c\), for the given function.

    13. \(f(x) =\sqrt{x},\quad n=4,\quad c=1\)

    14. \(f(x) =\ln (x+1),\quad n=4,\quad c=1\)

    15. \(f(x) =\cos x,\quad n=6,\quad c=\pi/4\)

    16. \(f(x) =\sin x,\quad n=5,\quad c=\pi/6\)

    17. \(f(x) =\frac{1}{x},\quad n=5,\quad c=2\)

    18. \(f(x) =\frac{1}{x^2},\quad n=8,\quad c=1\)

    19. \(f(x) =\frac{1}{x^2+1},\quad n=4,\quad c=-1\)

    20. \(f(x) =x^2 \cos x,\quad n=2,\quad c=-1\)

    In Exercises 21-24, approximate the function value with the indicated Taylor polynomial and give approximate bounds on the error.

    21. Approximate\(\sin 0.1\) with the Maclaurin polynomial of degree 3.

    22. Approximate \(\cos 1\) with the Maclaurin polynomial of degree 4.

    23. Approximate \(\sqrt{10}\) with the Taylor polynomial of degree 2 centered at \(x=9\).

    24. Approximate \(\ln 1.5\) with the Taylor polynomial of degree 3 centered at \(x=1\).

    Exercises 25-28 ask for an n to be found such that \(p_n(x)\) approximates \(f(x)\) within a certain bound of accuracy.

    25. Find n such that the Maclaurin polynomial of degree n of \(f(x)=e^x\) approximates within 0.0001 of the actual value.

    26. Find n such that the Taylor polynomial of degree n of \(f(x)=\sqrt{x}\), centered at \(x=4\), approximates \(\sqrt{3}\) within 0.0001 of the actual value.

    27. Find n such that the Maclaurin polynomial of degree n of \(f(x)=\cos x\) approximates \(\cos \pi/3\) within 0.0001 of the actual value.

    28. Find n such that the Maclaurin polynomial of degree n of \(f(x)=\sin x\) approximates \(\cos \pi\) within 0.0001 of the actual value.

    In Exercises 29-33, find the \(n^{th}\) term of the indicated Taylor polynomial.

    29. Find a formula for the \(n^{th}\) term of the Maclaurin polynomial for \(f(x)=e^x\).

    30. Find a formula for the \(n^{th}\) term of the Maclaurin polynomial for \(f(x)=\cos x\).

    31. Find a formula for the \(n^{th}\) term of the Maclaurin polynomial for \(f(x)=\frac{1}{1-x}\).

    32. Find a formula for the \(n^{th}\) term of the Maclaurin polynomial for \(f(x)=\frac{1}{1+x}\).

    33. Find a formula for the \(n^{th}\) term of the Maclaurin polynomial for \(f(x)=\ln x\).

    In Exercises 34-36, approximate the solution to the given differential equation with a degree 4 Maclaurin polynomial.

    34. \(y'=y,\quad y(0)=1\)

    35. \(y'=5y,\quad y(0)=3\)

    36. \(y'=\frac{2}{y},\quad y(0)=1\)

    8.8: Taylor Series

    Terms and Concepts

    1. What is the difference between a Taylor polynomial and a Taylor series?

    2 What theorem must we use to show that a function is equal to its Taylor series?

    Problems

    Key Idea 32 gives the \(n^{th}\) term of the Taylor series of common functions. In Exercises 3-6, verify the formula given in the Key Idea by finding the first few terms of the Taylor series of the given function and identifying a pattern.

    3. \(f(x) =e^x;\quad c=0\)

    4. \(f(x) =\sin x;\quad c=0\)

    5. \(f(x) =1/(1-x);\quad c=0\)

    6. \(f(x) =\tan^{-1} x;\quad c=0\)

    In Exercises 7-12, find a formula for the \(n^{th}\) term of the Taylor series of \(f(x)\), centered at c, by finding the coefficients of the first few powers of x and looking for a pattern. (The formulas for several of these are found in Key Idea 32; show work verifying these formula.)

    7. \(f(x) =\cos x;\quad c=\pi/2\)

    8. \(f(x) =1/x;\quad c=1\)

    9. \(f(x) =e^{-x};\quad c=\pi/2\)

    10. \(f(x) =\ln (1+x);\quad c=0\)

    11. \(f(x) =x/(x+1);\quad c=1\)

    12. \(f(x) =\sin x;\quad c=\pi/4\)

    In Exercises 13-16, show that the Taylor series for \(f(x)\), as given in Key Idea 32, is equal to \(f(x)\) by applying Theorem 77; that is, show \(\lim\limits_{n\to\infty}R_n(x)=0\).

    13. \(f(x) =e^x\)

    14. \(f(x) =\sin x\)

    15. \(f(x) =\ln x\)

    16. \(f(x) =1/(1-x)\) (show equality only on (-1,0))

    In Exercises 17-20, use the Taylor series given in Key Idea 32 to verify the given identity.

    17. \(\cos (-x)=\cos x\)

    18. \(\sin(-x)=-\sin x\)

    19. \(\frac{d}{dx}\left ( \sin x\right )=\cos x\)

    20. \(\frac{d}{dx}\left ( \cos x\right )=-\sin x\)

    In Exercises 21-24, write out the first 5 terms of the Binomial series with the given k-value.

    21. \(k=1/2\)

    22. \(k=-1/2\)

    23. \(k=1/3\)

    24. \(k=4\)

    In Exercises 25-30, use the Taylor series given in Key Idea 32 to create the Taylor series of the given functions.

    25. \(f(x)=\cos \left ( x^2\right )\)

    26. \(f(x)=e^{-x}\)

    27. \(f(x) =\sin (2x+3)\)

    28. \(f(x) =\tan^{-1}(x/2)\)

    29. \(f(x) =e^x \sin x\) (only find the first 4 terms).

    30. \(f(x)=(1+x)^{1/2}\cos x\) (only find the first 4 terms)

    In Exercises 31-32, approximate the value of the given definite integral by using the first 4 nonzero terms of the integrand's Taylor series.

    31. \(\int_0^{\sqrt{\pi}}\sin \left (x^2\right )\,dx\)

    32.\(\int_0^{\pi^2 /4}\cos \left (\sqrt{x}\right )\,dx\)