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
Solution
C
2. 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
Solution
D
3. 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. \( \displaystyle \left\{ \frac{n}{\ln{ n}} \right\}\)
Solution
E
4. 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.
Solution
A
5. 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. \( \displaystyle |r| > 1\)
Solution
C
6. 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
Solution
C
7. \( \displaystyle \sum u_n\) 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.
Solution
E
8. \( \displaystyle \sum_{n = 1}^{\infty} \frac{1}{n(n+1)} = \)
   1. \( \displaystyle \frac{4}{3} \)
   2. 1
   3. \( \displaystyle \frac{3}{2} \)
   4. \( \displaystyle \frac{3}{4} \)
   5. \( \displaystyle \infty\)
Solution
B
9. 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. \( \displaystyle \frac{3}{4}\)
Solution
A
10. 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
11. 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
12. 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
13. 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
14. 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
15. 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
16. 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
17. 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. \( \displaystyle \frac{1}{1 \cdot2} - \frac{1}{2 \cdot3} + \frac{1}{3 \cdot4} - \frac{1}{4 \cdot5} + ...\)
Solution
B
18. 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
19. 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
20. 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 \(x\)
Solution
A