# BC Calculus: Multiple Choice Worksheet 2 on Sequences and Series

• Class: 5H: BC Calculus
• Author: Peter Atlas
• Text: Calculus Finney, Demana, Waits, Kennedy

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
3. 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 $$\displaystyle x \neq -1$$
4. Solution A
5. 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. $$\displaystyle x < 2\text{ or }x > 4$$
6. Solution C
7. 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
8. Solution C
9. 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. $$\displaystyle -1 < x < 1$$
10. Solution D
11. 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. $$\displaystyle \frac{1}{384}$$
12. Solution E
13. 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. $$\displaystyle 1 - \frac{x}{2} + \frac{x^2}{4} - \frac{3x^3}{8}$$
14. Solution B
15. 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. $$\displaystyle e \left( 1 - (x - 1) + \frac{(x - 1)^2}{2}! - \frac{(x - 1)^3}{3!} \right)$$
16. Solution D
17. 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. $$\displaystyle -\frac{1}{3\sqrt{2}}$$
18. Solution D
19. 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.
20. Solution C
21. 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. $$\displaystyle \infty$$
22. Solution C
23. 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
24. Solution E
25. 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
26. Solution A
27. 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
28. Solution D
29. 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
30. Solution A