BC Calculus: Multiple Choice Worksheet 2 on Sequences and Series

Indicate the correct choice for each of the following:

  1. The power series \( \displaystyle x + \frac{x^2}{2} + \frac{x^3}{3} + ... + \frac{x^n}{n} + ... \) converges if and only if
    1. \( \displaystyle -1 < x < 1\)
    2. \( \displaystyle -1 \leq x \leq 1\)
    3. \( \displaystyle -1 \leq x < 1\)
    4. \( \displaystyle -1 < x \leq 1 \)
    5. \( \displaystyle x = 0\)
    2. Solution C
  2. The power series \( \displaystyle (x + 1) - \frac{(x + 1)^2}{2!}+ \frac{(x + 1)^3}{3!} - \frac{(x + 1)^4}{4!} + ... \) diverges
    1. for no real \( \displaystyle x\)
    2. if \( \displaystyle -2 < x \leq 0\)
    3. if \( \displaystyle x < -2\text{ or }x > 0\)
    4. if \( \displaystyle -2 \leq x < 0 \)
    5. if \( \displaystyle x \neq -1\)
    3. Solution A
  3. The series \( \displaystyle \sum_{n=1}^{\infty} n!(x - 3)^n\) converges if and only if
    1. \( \displaystyle x = 0\)
    2. \( \displaystyle 2 < x < 4\)
    3. \( \displaystyle x = 3\)
    4. \( \displaystyle 2 \leq x \leq 4\)
    5. \( \displaystyle x < 2\text{ or }x > 4\)
    4. Solution C
  4. The interval of convergence of the series obtained by differentiating term by term the series \( \displaystyle (x - 2) - \frac{(x - 2)^2}{4} + \frac{(x - 2)^3}{9} - \frac{(x -2)^4}{16} + ... \) is
    1. \( \displaystyle 1 \leq x \leq 3\)
    2. \( \displaystyle 1 \leq x < 3\)
    3. \( \displaystyle 1 < x \leq 3\)
    4. \( \displaystyle 0 \leq x \leq 4\)
    5. None of the preceding
    5. Solution C
  5. Let \( \displaystyle f(x) = \sum_{n=0}^{\infty} x^n\). The interval of convergence of \(\displaystyle \int_0^x f(t) \, dt \) is
    1. \( \displaystyle x = 0\) only
    2. \( \displaystyle |x| \leq 1\)
    3. \( \displaystyle -\infty < x < \infty\)
    4. \( \displaystyle -1 \leq x < 1\)
    5. \( \displaystyle -1 < x < 1\)
    6. Solution D
  6. The coefficient of \( \displaystyle x^4\) in the Maclaurin series for \( \displaystyle f(x) = e^{-\frac{x}{2}}\) is
    1. \( \displaystyle -\frac{1}{24}\)
    2. \( \displaystyle \frac{1}{24}\)
    3. \( \displaystyle \frac{1}{96}\)
    4. \( \displaystyle -\frac{1}{384}\)
    5. \( \displaystyle \frac{1}{384}\)
    7. Solution E
  7. The Maclaurin polynomial of order 3 for \( f(x) = \sqrt{1 + x}\) is
    1. \( \displaystyle 1 + \frac{x}{2} - \frac{x^2}{4} + \frac{3x^3}{8}\)
    2. \( \displaystyle 1 + \frac{x}{2} - \frac{x^2}{8}+ \frac{x^3}{16}\)
    3. \( \displaystyle 1 - \frac{x}{2} + \frac{x^2}{8}- \frac{x^3}{16}\)
    4. \( \displaystyle 1 + \frac{x}{2} - \frac{x^2}{8}+ \frac{x^3}{8}\)
    5. \( \displaystyle 1 - \frac{x}{2} + \frac{x^2}{4} - \frac{3x^3}{8}\)
    8. Solution B
  8. The Taylor polynomial of order 3 at \( \displaystyle x = 1\text{ for } e^x\) is
    1. \( \displaystyle 1 + (x - 1) + \frac{(x - 1)^2}{2} + \frac{(x - 1)^3}{3}\)
    2. \( \displaystyle e \left( 1 + (x - 1) + \frac{(x - 1)^2}{2} + \frac{(x - 1)^3}{3} \right) \)
    3. \( \displaystyle e \left( 1 + (x + 1) + \frac{(x + 1)^{2!}}{2} + \frac{(x + 1)^{3!}}{3!} \right)\)
    4. \( \displaystyle e \left( 1 + (x - 1) + \frac{(x - 1)^2}{2!} + \frac{(x - 1)^3}{3!} \right)\)
    5. \( \displaystyle e \left( 1 - (x - 1) + \frac{(x - 1)^2}{2}! - \frac{(x - 1)^3}{3!} \right)\)
    9. Solution D
  9. The coefficient of \( \displaystyle \left( x - \frac{\pi}{4} \right)^3\) in the Taylor series about \(\displaystyle \frac{\pi}{4}\) of \( \displaystyle f(x) = \cos{ x}\) is
    1. \( \displaystyle \frac{\sqrt{3}}{12} \)
    2. \( \displaystyle -\frac{1}{12} \)
    3. \( \displaystyle \frac{1}{12} \)
    4. \( \displaystyle \frac{1}{6\sqrt{2}} \)
    5. \( \displaystyle -\frac{1}{3\sqrt{2}}\)
    10. Solution D
  10. Which of the following series can be used to compute \( \ln {0.8}\)?
    1. \( \displaystyle \ln{(x - 1)}\) expanded about \( x = 0 \)
    2. \( \displaystyle \ln {x}\) about \(x = 0 \)
    3. \( \displaystyle \ln {x}\) in powers of \(x - 1 \)
    4. \( \displaystyle \ln{(x - 1)}\) in powers of \(x - 1 \)
    5. None of the preceding.
    11. Solution C
  11. The radius of convergence of the series \( \displaystyle \sum_{n = 1}^{\infty} \frac{x^n}{2^n} \cdot \frac{n^n}{n!}\) is
    1. 0
    2. 2
    3. \( \displaystyle \frac{2}{e} \)
    4. \( \displaystyle \frac{e}{2} \)
    5. \( \displaystyle \infty\)
    12. Solution C
  12. If the approximate formula \( \displaystyle \sin{x} = x - \frac{x^3}{3!}\) is used and \(|x| < 1\), then the error is numerically less than
    1. 0.001
    2. 0.003
    3. 0.005
    4. 0.008
    5. 0.009
    13. Solution E
  13. If an appropriate series is used to evaluate \( \displaystyle \int_0^{0.3} x^2 e^{-x^2} \, dx \), then, correct to three decimal places, the definite integral equals
    1. 0.009
    2. 0.082
    3. 0.098
    4. 0.008
    5. 0.090
    14. Solution A
  14. Note: The following is a calculator active question. The question can be done without a calculator, but it is very, very difficult! The sum of the series \( \displaystyle \sum_{n = 1}^{\infty} \left( \frac{\pi^3}{3^{\pi}} \right) ^n = \)
    1. 0
    2. 1
    3. \( \displaystyle \frac{3^{\pi}}{\pi^3 - 3^{\pi}} \)
    4. \( \displaystyle \frac{\pi^3}{3^{\pi} - \pi^3} \)
    5. None of these
    15. Solution D
  15. When \( \displaystyle \sum_{n = 1}^{\infty} (-1)^{n - 1} \frac{1}{3n - 1}\) is approximated by the sum of its first 300 terms, the error is closest to
    1. 0.001
    2. 0.002
    3. 0.005
    4. 0.010
    5. 0.020
    16. Solution A