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Section Practice

These are practice problems for the tests. Solutions are in the next section. While we will not present these, I am happy to answer questions about them in class.

Sequences

  1. Write the first five terms of the sequence in simplest form.

    1. \(\dsp \{5 - {{2n} \over 3} \}_{n=2}^{\infty}\)

    2. \(\dsp \{\cos(\frac{n \pi}{4}) \}_{n=0}^{\infty}\)

    3. \(\dsp \{{{3^n + 2} \over {2^n}} \}_{n=1}^{\infty}\)

  2. Write a formula for each sequence. I.e. \(x_n = \_\_\_\) for \(n = 1,2,3,\dots\text{.}\)

    1. \(\dsp {3 \over 5}, {6 \over {25}}, {9 \over {125}}, {{12} \over {625}}, \cdots\)

    2. \(\dsp -1, 1, -1, 1, -1, \cdots\)

    3. \(\dsp {1 \over 2}, -{2 \over 3}, {3 \over 4}, -{4 \over 5}, {5 \over 6}, \cdots\)

  3. What is the least upper bound and the greatest lower bound of each given sequence? Prove it.

    1. \(\dsp \{ \frac{2n-2}{n} \}_{n=1}^\infty\)

    2. \(\dsp \{ \frac{2n+5}{3n+1} \}_{n=2}^\infty\)

  4. Is the sequence monotonic or not? Prove it.

    1. \(\dsp \{ \frac{n}{n^2+1} \}_{n=1}^\infty\)

    2. \(\dsp \{\sin(\frac{2n \pi}{3}) \}_{n=1}^{\infty}\)

    3. \(\dsp \{\frac{n-2}{n} \}_{n=1}^{\infty}\)

  5. Use limits to find whether or not the sequence converges.

    1. \(\dsp x_n = {{n+2} \over {n^2+3n+2}}\)

    2. \(\dsp x_n = n \sin \Big({1 \over n} \Big)\)

    3. \(\dsp x_n = \Big({{1} \over {n}} \Big)^n\)

    4. \(\dsp x_n = \Big(1 - {1 \over n} \Big)^n\) Remember, the natural log function \(\ln\) is your friend.

  6. Use the definition of limit to find the smallest \(N\) so that \(\dsp \Big|{3 \over n}-0 \Big| \lt .001\) for \(n \ge N\text{.}\)

  7. Assume \(\epsilon\) is a small positive number and use the definition of limit to find the smallest \(N\) in terms of \(\epsilon\) so that \(\dsp \Big|{{n+3} \over {2n}}- {1 \over 2} \Big| \lt \epsilon\) for \(n \ge N\text{.}\)

Series

  1. List, in simplest form, the first three partial sums for the series and write a formula for the \(N^{th}\) partial sum.

    1. \(\dsp 1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} + \dots\)

    2. \(\dsp 3 + \frac{3}{5} + \frac{3}{25} + \frac{3}{125} + \dots\)

  2. What does the geometric series converge to?

    1. \(\dsp \sum_{n=1}^{\infty} {3 \over {4^n}}\)

    2. \(\dsp \sum_{n=1}^{\infty} \Big({9 \over {10}} \Big)^n\)

  3. Use the fact that the Harmonic Series diverges to argue that \(\dsp \sum_{n=3}^\infty \frac{5}{3n}\) diverges.

  4. Apply the integral test to determine if the series converges.

    1. \(\dsp \sum_{n=2}^{\infty} {{2n} \over {(n^2-3)^2}}\)

    2. \(\dsp \sum_{n=2}^{\infty} {{\ln(n)} \over {n}}\)

  5. Determine if these series converge via one of the \(n^{th}\) term test, integral test, p-series test, comparison test, or limit comparison test.

    1. \(\dsp \sum_{n=2}^{\infty} {2 \over {1-n^2}}\)

    2. \(\dsp \sum_{n=1}^{\infty} {1 \over {4n^2+2n}}\)

    3. \(\dsp \sum_{n=0}^{\infty} \Big({{n+1} \over {n+3}}\Big)\)

    4. \(\dsp \sum_{n=1}^{\infty} \Big(\sin \Big({{\pi} \over 4} \Big) \Big)^n\)

    5. \(\dsp \sum_{n=0}^{\infty} {{n+2} \over {3n+4}}\)

    6. \(\dsp \sum_{n=0}^{\infty} {1 \over {3n}} + {1 \over {2^n}}\)

    7. \(\dsp \sum_{n=1}^{\infty} {3 \over {2^n}} + {4 \over {3^n}}\)

    8. \(\dsp \sum_{n=2}^{\infty} 4+{1 \over {n^2}}\)

    9. \(\dsp \sum_{n=1}^{\infty} {1 \over n}- {1 \over {n+2}}\)

    10. \(\dsp \sum_{n=0}^{\infty} {e^{-5n}}\)

    11. \(\dsp \sum_{n=1}^{\infty} {{3n} \over {n^4 + 16}}\)

    12. \(\dsp \sum_{n=3}^{\infty} \sqrt{{1} \over {3n+4}}\)

    13. \(\dsp \sum_{n=2}^{\infty} {{2} \over {n^2-1}}\)

    14. \(\dsp \sum_{n=1}^{\infty} {1 \over {n^n}}\)

    15. \(\dsp \sum_{n=1}^{\infty} {2 \over {\sqrt{n^2+2n}}}\)

    16. \(\dsp \sum_{n=1}^{\infty} {{2} \over {n3^n}}\)

    17. \(\dsp \sum_{n=2}^{\infty} {1 \over {n^2+5n+6}}\)

    18. \(\dsp \sum_{n=2}^{\infty} {1 \over {3^n-1}}\)

    19. \(\dsp \sum_{n=2}^{\infty} {{\ln(n)} \over {n^3}}\)

    20. \(\dsp \sum_{n=2}^{\infty} {{\ln(n^2)} \over {n^2}}\)

  6. Determine whether the alternating series is convergent or divergent.

    1. \(\dsp \sum_{n=2}^{\infty} (-1)^n{{2} \over {n^2-1}}\)

    2. \(\dsp \sum_{n=0}^{\infty} (-1)^n {{3n} \over {4^n}}\)

    3. \(\dsp \sum_{n=1}^{\infty} (-1)^n{{4^n} \over {n^2}}\)

    4. \(\dsp \sum_{n=3}^{\infty} {{(-5)^{n-2}} \over {3^{n+1}}}\)

    5. \(\dsp \sum_{n=2}^{\infty} (-1)^n \Big( {1 \over {\log(n)}} \Big)\)

    6. \(\dsp \sum_{n=0}^{\infty} (-1)^n \Big({{3} \over {n+3}}\Big)\)

    7. \(\dsp \sum_{n=0}^{\infty} \cos(n \pi)\)

    8. \(\dsp \sum_{n=1}^{\infty} (-1)^n{{3^n} \over {3n}}\)

    9. \(\dsp \sum_{n=1}^{\infty} (-1)^n \Big({1\over {\sqrt n}} \Big)\)

    10. \(\dsp \sum_{n=1}^{\infty} (-1)^n{3 \over {n^{10}}}\)

    11. \(\dsp \sum_{n=2}^{\infty} (-1)^n\Big(4 + {{1} \over {n^2}} \Big)\)

    12. \(\dsp\sum_{n=1}^{\infty} {{\sec(n \pi)} \over {n}}\)

  7. Determine if it converges absolutely,converges conditionally, or diverges.

    1. \(\dsp \sum_{n=2}^{\infty} {{(-4)^n} \over {3^{n+1}}}\)

    2. \(\dsp \sum_{n=2}^{\infty} (-1)^n{2 \over {n-1}}\)

    3. \(\dsp \sum_{n=2}^{\infty} (-1)^n{{1} \over {n(\log_4 n)^2}}\)

    4. \(\dsp \sum_{n=2}^{\infty} \Big({{-5} \over {8}}\Big)^n\)

    5. \(\dsp \sum_{n=1}^{\infty} (-1)^n {{n^n} \over{n!}}\)

    6. \(\dsp \sum_{n=2}^{\infty} (-1)^n {{n!} \over{n^n}}\)

    7. \(\dsp \sum_{n=1}^{\infty} \cos(n \pi)\)

    8. \(\dsp \sum_{n=2}^{\infty} (-1)^n{{3n} \over {n^2-1}}\)

    9. \(\dsp \sum_{n=1}^{\infty} {{3n} \over {(-3)^n}}\)

    10. \(\dsp\sum_{n=1}^{\infty} (-1)^n {{\sqrt{n}} \over {n+4}}\)

    11. \(\dsp \sum_{n=1}^{\infty} {{n^2} \over {(-2)^n}}\)

    12. \(\dsp \sum_{n=1}^{\infty} {{n^e} \over {(-e)^n}}\)

  8. For each of the following power series, determine its interval of convergence.

    1. \(\dsp \sum_{n=1}^{\infty} {{x^n} \over {n+3}}\)

    2. \(\dsp \sum_{n=0}^{\infty} (-x)^n{1 \over {4^n}}\)

    3. \(\dsp \sum_{n=1}^{\infty} (-1)^n{{3^n} \over {n^2}}x^n\)

    4. \(\dsp \sum_{n=0}^{\infty} {{(x-5)^n} \over {3^{n+1}}}\)

    5. \(\dsp \sum_{n=1}^{\infty} {{x^n} \over {\sqrt{n^2+1}}}\)

    6. \(\dsp \sum_{n=1}^{\infty} {{x^n} \over {\log(n+1)}}\)

    7. \(\dsp \sum_{n=0}^{\infty} {{(x+3)^n}\over{n^n}}\)

    8. \(\dsp \sum_{n=0}^{\infty} {{\cos(n \pi)x^n}\over {n!}}\)

    9. \(\dsp \sum_{n=1}^{\infty} {{(x+4)^n} \over{(n+4)4^n}}\)

    10. \(\dsp \sum_{n=1}^{\infty} \Big(1+ {1 \over n} \Big)^{2n} x^n\)

    11. \(\dsp \sum_{n=0}^{\infty} n^{2n}x^n\)

    12. \(\dsp \sum_{n=1}^{\infty} {{(x-2)^n} \over {2n+2}}\)

  9. For each of the following functions, find its Taylor or MacLaurin series and state the radius of convergence.

    1. \(\dsp f(x) = -{1 \over {x^2}}\text{;}\) \(c=1\)

    2. \(\dsp f(x) = \sin(2x)\text{;}\) \(c = 0\)

    3. \(\dsp f(x) = 3^x\text{;}\) \(c = 1\)

    4. \(\dsp f(x) = e^x\text{;}\) \(c = 2\)

    5. \(\dsp f(x) = \cos(x)\text{;}\) \(c= {{\pi} \over 2}\)

    6. \(\dsp f(x) = \ln(1+x)\text{;}\) \(c = 0\)

  10. Use multiplication, substitution, differentiation, and integration whenever possible to find a series representation of each of the following functions.

    1. \(\dsp f(x) =x^2 \cos(x)\) Multiply \(x^2\) times the Taylor series for cosine.

    2. \(\dsp f(x) ={{\sin(x^2)} \over x}\)

    3. \(\dsp f(x) = 5xe^{3x}\)

    4. \(\dsp f(x) = {2 \over {(1-x)^2}}\)

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