# BC Calculus: Multiple Choice Worksheet 1 on Sequences and Series

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

NB: These problems should be done without a calculator.

Indicate the correct choice for each of the following:

1. If $$\displaystyle \{s_n\} = \left\{ 1 + \frac{(-1)^n}{n} \right\}$$, then
1. $$\displaystyle \{s_n\}$$ diverges by oscillation
2. $$\displaystyle \{s_n\}$$ converges to zero
3. $$\displaystyle \lim_{n \to \infty} s_n = 1$$
4. $$\displaystyle \{s_n\}$$ diverges to infinity
5. None of the above is true
2. Solution C
3. The sequence $$\left\{ \sin{ \frac{n\pi}{6}} \right\}$$
1. is unbounded
2. is monotonic
3. converges to a number less than 1
4. is bounded
5. diverges to infinity
4. Solution D
5. Which of the following sequences diverges?
1. $$\displaystyle \left\{ \frac{1}{n} \right\}$$
2. $$\displaystyle \left\{ \frac{(-1)^{n + 1}}{n} \right\}$$
3. $$\displaystyle \left\{ \frac{2^n}{e^n} \right\}$$
4. $$\displaystyle \left\{ \frac{n^2}{e^n} \right\}$$
5. $$\displaystyle \left\{ \frac{n}{\ln{ n}} \right\}$$
6. Solution E
7. Which of the following statements about sequences is false?
1. If $$\displaystyle \{s_n\}$$ is bounded, then it is convergent.
2. $$\displaystyle \lim_{n \to \infty} s_n = L \implies |s_n - L| < 0.001$$ except for at most a finite number of $$n$$'s.
3. If $$\displaystyle \{s_n\}$$ converges, then $$\displaystyle \{s_n\}$$is bounded.
4. If $$\displaystyle \{s_n\}$$ is unbounded, then it diverges.
5. None of the above.
8. Solution A
9. The sequence $$\displaystyle \{r^n \}$$ converges if and only if
1. $$\displaystyle |r| < 1$$
2. $$|r| \leq 1$$
3. $$\displaystyle -1 < r \leq 1$$
4. $$\displaystyle 0 < r < 1$$
5. $$\displaystyle |r| > 1$$
10. Solution C
11. The sequence $$\displaystyle \{s_n\}$$ , where $$\displaystyle s_n = \frac{n}{n + 1}$$, converges to 1. It follows then, if $$\displaystyle \epsilon > 0$$, that there exists a positive integer $$N$$ such that $$\displaystyle n > N \implies \left| s_n - 1 \right| < \epsilon$$. Let $$\epsilon = 0.01$$; then the least such $$N$$ is
1. 10
2. 90
3. 99
4. 100
5. 101
12. Solution C
13. $$\displaystyle \sum u_n$$ is a series of constants for which $$\displaystyle \lim_{n \to \infty} u_n = 0$$. Which of the following statements is always true?
1. $$\displaystyle \sum u_n$$ converges to a finite sum.
2. $$\displaystyle \sum u_n = 0.$$
3. $$\displaystyle \sum u_n$$ does not diverge to infinity.
4. $$\displaystyle \sum u_n$$ is a positive series.
5. None of the preceding.
14. Solution E
15. $$\displaystyle \sum_{n = 1}^{\infty} \frac{1}{n(n+1)} =$$
1. $$\displaystyle \frac{4}{3}$$
2. 1
3. $$\displaystyle \frac{3}{2}$$
4. $$\displaystyle \frac{3}{4}$$
5. $$\displaystyle \infty$$
16. Solution B
17. The sum of the series $$\left( 2 - 1 + \frac{1}{2} - \frac{1}{4} + \frac{1}{8} - ... \right)$$ is
1. $$\displaystyle \frac{4}{3}$$
2. $$\displaystyle \frac{5}{4}$$
3. 1
4. $$\displaystyle \frac{3}{2}$$
5. $$\displaystyle \frac{3}{4}$$
18. Solution A
19. Which of the following statements about series is true?
1. $$\displaystyle \lim_{n \to \infty} u_n = 0, \implies \sum u_n$$ converges.
2. $$\displaystyle \lim_{n \to \infty} u_n \neq 0 \implies \sum u_n$$ converges.
3. $$\displaystyle \sum u_n$$ diverges $$\displaystyle \implies \lim_{n \to \infty} u_n \neq 0 .$$
4. $$\displaystyle \sum u_n$$ converges $$\displaystyle \iff \lim_{n \to \infty} u_n = 0.$$
5. None of the preceding.
Solution E
20. Which of the following statements about series is false?
1. $$\displaystyle \sum_{k = 1}^{\infty} u_k = \sum_{k = m}^{\infty} u_k$$, where $$m$$ is any positive integer.
2. If $$\displaystyle \sum u_n$$ converges, so does $$\displaystyle \sum c u_n$$, if $$c \neq 0.$$
3. If $$\displaystyle \sum a_n$$ and $$\displaystyle \sum b_n$$ converge, so does $$\displaystyle \sum (c a_n + b_n)$$, where $$c \neq 0.$$
4. If 1000 terms are added to a convergent series, the new series also converges.
5. Rearranging the terms of a positive convergent series will not affect its convergence or its sum.
Solution A
21. Which of the following series converges?
1. $$\displaystyle \sum_{n=1}^{\infty} \frac{1}{\sqrt[3]{n}}$$
2. $$\displaystyle \sum_{n=1}^{\infty} \frac{1}{ \sqrt{n}}$$
3. $$\displaystyle \sum_{n=1}^{\infty} \frac{1}{n}$$
4. $$\displaystyle \sum_{n=1}^{\infty} \frac{1}{10n - 1}$$
5. $$\displaystyle \sum_{n=1}^{\infty} \frac{2}{n^2 - 5}$$
Solution E
22. Which of the following series diverges?
1. $$\displaystyle \sum_{n = 1}^{\infty} \frac{1}{n(n + 1)}$$
2. $$\displaystyle \sum_{n = 1}^{\infty} \frac{n + 1}{n!}$$
3. $$\displaystyle \sum _{n = 2}^{\infty} \frac{1}{n \ln{ n}}$$
4. $$\displaystyle \sum_{n = 1}^{\infty} \frac{ \ln{ n}}{2^n}$$
5. $$\displaystyle \sum_{n = 1}^{\infty} \frac{n }{2^n}$$
Solution C
23. Which of the following series diverges?
1. $$\displaystyle \sum \frac{1}{n^2}$$
2. $$\displaystyle \sum \frac{1}{n^2 + n}$$
3. $$\displaystyle \sum \frac{n }{n^3 + 1}$$
4. $$\displaystyle \sum \frac{n }{\sqrt{4n^2 - 1}}$$
5. none of the preceding.
Solution D
24. For which of the following series does the Ratio Test fail?
1. $$\displaystyle \sum \frac{1}{n!}$$
2. $$\displaystyle \sum \frac{n}{2^n}$$
3. $$\displaystyle 1 + \frac{1}{2^{\frac{3}{2}}} + \frac{1}{3^{\frac{3}{2}}} + \frac{1}{4^{\frac{3}{2}}} + ...$$
4. $$\displaystyle \frac{\ln{ 2}}{2^2} + \frac{\ln{ 3}}{2^3} + \frac{\ln{ 4}}{2^4} + ...$$
5. $$\displaystyle \sum \frac{n^n}{n!}$$
Solution C
25. Which of the following alternating series diverges?
1. $$\displaystyle \sum (-1)^{n-1} \frac{1}{n}$$
2. $$\displaystyle \sum (-1)^{n + 1}\frac{n - 1}{n + 1}$$
3. $$\displaystyle \sum (-1)^{n + 1}\frac{1}{\ln{ (n + 1)}}$$
4. $$\displaystyle \sum (-1)^{n-1}\frac{1}{\sqrt{n}}$$
5. $$\displaystyle \sum (-1)^{n-1}\frac{n}{n^2 + 1}$$
Solution B
26. Which of the following series converges conditionally?
1. $$\displaystyle 3 - 1 + \frac{1}{3} - \frac{1}{9} + ...$$
2. $$\displaystyle \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{4}} - ...$$
3. $$\displaystyle \frac{1}{2^2} - \frac{1}{3^2} + \frac{1}{4^2} - ...$$
4. 1 - 1.1 + 1.21 - 1.331 + ...
5. $$\displaystyle \frac{1}{1 \cdot2} - \frac{1}{2 \cdot3} + \frac{1}{3 \cdot4} - \frac{1}{4 \cdot5} + ...$$
27. Solution B
28. Let S =$$\displaystyle \sum_{n = 1}^{\infty} \left( \frac{2}{3} \right)^n$$; then $$S =$$
1. 1
2. $$\displaystyle \frac{3}{2}$$
3. $$\displaystyle \frac{4}{3}$$
4. 2
5. 3
Solution D
29. Which of the following statements is true?
1. If a series converges, then it converges absolutely.
2. If a series is truncated after the $$n$$th term, then the error is less than the first term ommitted.
3. If the first terms of an alternating series decrease, then the series converges.
4. If $$r < 1$$, then $$\displaystyle \sum r^n$$ converges.
5. None of the preceding.
Solution E
30. Which of the following expansions is impossible?
1. $$\displaystyle \sqrt{x - 1}$$ in powers of $$x$$
2. $$\sqrt{x + 1}$$ in powers of $$x$$
3. $$\displaystyle \ln{ x}$$ in powers of $$x - 1$$
4. $$\displaystyle \tan {x}$$ in powers of $$x - \frac{\pi}{4}$$
5. $$\displaystyle \ln{ (1 - x)}$$ in powers of $$x$$
31. Solution A