AP Calculus BC Unit 10: Infinite Sequences and Series (BC ONLY) Practice Exam

Assessment for Unit 10: Infinite Sequences and Series (BC ONLY)

Estimated Time: 60 minutes

Exam Details

Questions: 48 total (45 multiple choice, 3 short answer)
Time: 60 minutes (recommended)
Topics Covered: Unit 10: Infinite Sequences and Series (BC ONLY)
Format: Multiple Choice, Short Answer questions matching real exam difficulty
Scoring: Instant results with detailed explanations

A. Multiple-Choice Questions (45 Questions)

Select the one best answer for each question.

1. The graph shows the first six terms of the sequence of partial sums, $S_n = \sum_{k=1}^n a_k$, for an infinite series. The graph of the sequence $S_n$ is strictly increasing and has a horizontal asymptote at $y = 3$. Based on the graph, which of the following statements is true about the infinite series $\sum_{k=1}^{\infty} a_k$?

(A)The series diverges.(B)The series converges to 3.(C)The series converges to a value greater than 3.(D)The series converges, but the sum cannot be determined without the explicit formula for $a_k$.

2. The $n$th partial sum of the infinite series $\sum_{k=1}^{\infty} a_k$ is given by the formula $S_n = \frac{8n^2 + 2n}{2n^2 + 5}$. What is the value of $\sum_{k=1}^{\infty} a_k$?

(A)4(B)8(C)0(D)The series diverges.

3. Which of the following statements correctly describes the behavior of the series $\sum_{n=1}^{\infty} \left( \frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n+1}} \right)$?

(A)The series diverges because the limit of the terms is 0.(B)The series diverges because the partial sums grow without bound.(C)The series converges to 0.(D)The series converges to 1.

4. Let $S$ be the sum of the alternating series $\sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{n^2 + 3}$. If the partial sum $S_3 = \frac{1}{4} - \frac{1}{7} + \frac{1}{12}$ is used to approximate $S$, which of the following is the alternating series error bound?

(A)$\frac{1}{12}$(B)$\frac{1}{19}$(C)$\frac{1}{28}$(D)$\frac{1}{39}$

5. The series $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sqrt{n+1}}$ converges. What is the least number of terms of the series that must be summed to guarantee that the alternating series error bound is less than $0.2$?

(A)23(B)24(C)25(D)26

6. Let $f$ be a function having derivatives of all orders for all real numbers. If $f(3) = 2$, $f'(3) = -4$, $f''(3) = 6$, and $f'''(3) = 12$, which of the following is the third-degree Taylor polynomial for $f$ about $x = 3$?

(A)$2 - 4(x-3) + 6(x-3)^2 + 12(x-3)^3$(B)$2 - 4(x-3) + 3(x-3)^2 + 2(x-3)^3$(C)$2 - 4(x+3) + 3(x+3)^2 + 2(x+3)^3$(D)$2 - 4x + 3x^2 + 2x^3$

7. The function $f$ is approximated near $x=0$ by the third-degree Taylor polynomial $P_3(x) = 4 - 3x + \frac{x^2}{2} - 5x^3$. What is the value of $f'''(0)$?

(A)-30(B)-5(C)-5/6(D)5

8. Let $f$ be the function given by $f(x) = \sqrt{x}$. What is the approximation for $f(4.2)$ found by using the second-degree Taylor polynomial for $f$ about $x = 4$?

(A)2.049(B)2.050(C)2.051(D)2.100

9. Let f be a function having derivatives of all orders for all real numbers. The table below gives the maximum values of the absolute value of the k -th derivative of f on the closed interval [2, 2.5] .$\n\n$| k | $\max_{2 \le x \le 2.5} $|f^{(k)}(x)| |$\n$|:---:|:---:|$\n$| 1 | 8 |$\n$| 2 | 12 |$\n$| 3 | 24 |$\n$| 4 | 60 |$\n\nLet $ P_2(x) be the second-degree Taylor polynomial for f about x = 2 . Using the Lagrange error bound, which of the following is the smallest value B that guarantees |f(2.5) - P_2(2.5)| $\le $B ?

(A) $\frac{1}{2} $(B) $\frac{1}{4} $(C) $\frac{1}{8} $(D) $\frac{5}{16} $

10. Let f be a function such that |f^{(k)}(x)| $\le $5 for all x and all positive integers k . Let P_n(x) be the n -th degree Taylor polynomial for f about x=0 . What is the least integer n for which the Lagrange error bound guarantees that |f(1) - P_n(1)| $\le \frac{1}{100} $ ?

(A)4(B)5(C)6(D)7

11. Which of the following is the interval of convergence for the power series $\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^n (x+2)^n}{\sqrt{n} \cdot 2^n}$ ?

(A)$(-4, 0]$(B)$[-4, 0)$(C)$(-4, 0)$(D)$[-4, 0]$

12. Let $f$ be the function defined by the power series $f(x) = \displaystyle \sum_{n=1}^{\infty} \frac{(x-1)^n}{n \cdot 5^n}$. Which of the following is the interval of convergence for the power series $f'(x)$ ?

(A)$[-4, 6)$(B)$(-4, 6]$(C)$(-4, 6)$(D)$[-4, 6]$

13. Consider the power series $\displaystyle \sum_{n=0}^{\infty} a_n (x-3)^n$. If $\displaystyle \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 4$, what is the radius of convergence of the series?

(A)$4$(B)$\frac{1}{4}$(C)$3$(D)$\infty$

14. Let $f$ be a function having derivatives of all orders for all real numbers. If $f(2) = 5$, $f'(2) = -3$, $f''(2) = 8$, and $f'''(2) = 12$, which of the following is the third-degree Taylor polynomial for $f$ about $x = 2$?

(A)$P_3(x) = 5 - 3(x-2) + 8(x-2)^2 + 12(x-2)^3$(B)$P_3(x) = 5 - 3(x-2) + 4(x-2)^2 + 2(x-2)^3$(C)$P_3(x) = 5 - 3(x+2) + 4(x+2)^2 + 2(x+2)^3$(D)$P_3(x) = 5 - 3x + 4x^2 + 2x^3$

15. Which of the following is the Maclaurin series for the function $f(x) = \frac{2x}{1 + 3x}$ ?

(A)$\sum_{n=0}^{\infty} 2 \cdot 3^n x^{n+1}$(B)$\sum_{n=0}^{\infty} (-1)^n 2 \cdot 3^n x^{n+1}$(C)$\sum_{n=0}^{\infty} (-1)^n 6^n x^{n+1}$(D)$\sum_{n=0}^{\infty} (-1)^n 2 \cdot 3^n x^n$

16. The Maclaurin series for $\sin x$ is $\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$. Which of the following is the general term for the Maclaurin series for $g(x) = x \sin(2x^2)$ ?

(A)$(-1)^n \frac{2^{2n+1} x^{4n+3}}{(2n+1)!}$(B)$(-1)^n \frac{2 x^{4n+3}}{(2n+1)!}$(C)$(-1)^n \frac{2^{2n+1} x^{2n+3}}{(2n+1)!}$(D)$(-1)^n \frac{4^{n} x^{4n+2}}{(2n+1)!}$

17. Which of the following is the Maclaurin series for the function $f$ defined by $f(x) = \frac{x}{2+x^2}$?

(A)$\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{2^{n+1}}$(B)$\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{2^{n+1}}$(C)$\sum_{n=0}^{\infty} \frac{x^{2n+1}}{2^{n+1}}$(D)$\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{2^n}$

18. Which of the following is the power series expansion for $f(x) = \frac{1}{(1-2x)^2}$ centered at $x=0$?

(A)$\sum_{n=0}^{\infty} (2x)^{2n}$(B)$\sum_{n=0}^{\infty} (n+1)(2x)^n$(C)$\sum_{n=0}^{\infty} n(2x)^{n-1}$(D)$\sum_{n=0}^{\infty} \frac{(2x)^{n+1}}{n+1}$

19. The Maclaurin series for $\ln(1+x)$ is $\sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n}$. Which of the following is the Maclaurin series for $f(x) = x \ln\left(1 + \frac{x}{3}\right)$?

(A)$\sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^{n+1}}{3^n n}$(B)$\sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^{n+1}}{3^{n+1} n}$(C)$\sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^{n}}{3^n n}$(D)$\sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^{n+2}}{3^n (n+1)}$

20. Which of the following gives the sum of the series $\sum_{n=1}^{\infty} \frac{4^{n+1}}{5^n}$ ?

(A)16(B)20(C)80(D)The series diverges.

21. For which values of $x$ does the infinite series $\sum_{n=0}^{\infty} \left( \frac{3x - 2}{4} \right)^n$ converge?

(A)$-2 < x < 6$(B)$-\frac{2}{3} < x < 2$(C)$-\frac{2}{3} \le x \le 2$(D)$x < 2$

22. Consider the infinite series $6 - 4 + \frac{8}{3} - \frac{16}{9} + \dots$. Which of the following statements is true?

(A)The series diverges because the terms alternate in sign.(B)The series diverges because $|r| \ge 1$.(C)The series converges to $\frac{18}{5}$.(D)The series converges to $18$.

23. The first four terms of a sequence $a_n$ are given by the table below. Assuming the behavior of the terms continues consistent with the trend shown, which of the following describes the behavior of the infinite series $\sum_{n=1}^{\infty} a_n$?$\n\n$| $n$ | 1 | 2 | 3 | 4 |$\n$| :--- | :--- | :--- | :--- | :--- |$\n$| $a_n$ | 1.5 | 1.25 | 1.125 | 1.0625 |

(A)The series diverges because $\lim_{n \to \infty} a_n = 1 \neq 0$.(B)The series converges because $\lim_{n \to \infty} a_n = 1$.(C)The series diverges because the terms are decreasing.(D)The $n^{th}$ term test is inconclusive because $\lim_{n \to \infty} a_n \neq 0$.

24. Which of the following infinite series diverges specifically as a result of the $n^{th}$ Term Test for Divergence?

(A)$\sum_{n=1}^{\infty} \frac{1}{n}$(B)$\sum_{n=1}^{\infty} \frac{1}{e^n}$(C)$\sum_{n=1}^{\infty} \frac{n}{2n+1}$(D)$\sum_{n=1}^{\infty} \frac{1}{n^2}$

25. If the infinite series $\sum_{n=1}^{\infty} a_n$ converges to a finite sum $S$, which of the following statements must be true?

(A)$\lim_{n \to \infty} a_n = S$(B)$\lim_{n \to \infty} a_n = 0$(C)$\sum_{n=1}^{\infty} |a_n|$ converges.(D)The series converges by the Alternating Series Test.

26. Let $f$ be a function that is continuous, positive, and decreasing for all $x \ge 1$, such that $a_n = f(n)$ for all integers $n \ge 1$. Based on the Integral Test, which of the following statements correctly describes the relationship between the series $\sum_{n=1}^{\infty} a_n$ and the improper integral $\int_{1}^{\infty} f(x) \, dx$?

(A)If $\int_{1}^{\infty} f(x) \, dx = L$, then $\sum_{n=1}^{\infty} a_n = L$.(B)If $\int_{1}^{\infty} f(x) \, dx$ diverges, then $\sum_{n=1}^{\infty} a_n$ converges to 0.(C)The series $\sum_{n=1}^{\infty} a_n$ converges if and only if $\int_{1}^{\infty} f(x) \, dx$ converges.(D)The series $\sum_{n=1}^{\infty} a_n$ converges if and only if $\lim_{x \to \infty} f(x) = 0$.

27. Consider the infinite series $\sum_{n=2}^{\infty} \frac{1}{n \ln(n)}$. Which of the following correctly determines the convergence or divergence of the series using the Integral Test?

(A)The series converges because $\int_{2}^{\infty} \frac{1}{x \ln(x)} \, dx$ converges to $\ln(\ln(2))$.(B)The series converges because $\lim_{n \to \infty} \frac{1}{n \ln(n)} = 0$.(C)The series diverges because $\int_{2}^{\infty} \frac{1}{x \ln(x)} \, dx$ diverges.(D)The series diverges because the function $f(x) = \frac{1}{x \ln(x)}$ is not decreasing for $x \ge 2$.

28. Which of the following statements is true regarding the convergence of the series $\sum_{n=1}^{\infty} n e^{-n^2}$ ?

(A)The series diverges by the nth Term Test for Divergence.(B)The series diverges because $\int_{1}^{\infty} x e^{-x^2} \, dx$ is infinite.(C)The series converges because $\int_{1}^{\infty} x e^{-x^2} \, dx = \frac{1}{2e}$.(D)The series converges because $\int_{1}^{\infty} x e^{-x^2} \, dx = \frac{1}{e}$.

29. Which of the following statements correctly describes the convergence behavior of the series $\sum_{n=1}^{\infty} \frac{1}{n}$ and $\sum_{n=1}^{\infty} \frac{1}{n^{1.01}}$?

(A)Both series diverge.(B)Both series converge.(C)The series $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges, while the series $\sum_{n=1}^{\infty} \frac{1}{n^{1.01}}$ converges.(D)The series $\sum_{n=1}^{\infty} \frac{1}{n}$ converges, while the series $\sum_{n=1}^{\infty} \frac{1}{n^{1.01}}$ diverges.

30. Consider the series defined by $\sum_{n=1}^{\infty} \frac{1}{n^{3k-2}}$, where $k$ is a constant. For which values of $k$ does the series converge?

(A)k > 1(B)k > $\frac{2}{3}$(C)k $\ge $1(D)k > $\frac{1}{3}$

31. Which of the following series diverge? I. $\sum_{n=1}^{\infty} \frac{1}{n^{\pi/e}}$ II. $\sum_{n=1}^{\infty} \frac{1}{\sqrt[3]{n^2}}$ III. $\sum_{n=1}^{\infty} \frac{1}{n^{\ln 2}}$

(A)II only(B)III only(C)I and II only(D)II and III only

32. Which of the following series converges?

(A)$\sum_{n=1}^{\infty} \frac{n}{n+2}$(B)$\sum_{n=1}^{\infty} \frac{n^2}{n^3+1}$(C)$\sum_{n=1}^{\infty} \frac{1}{3^n+n}$(D)$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$

33. Consider the series $\sum_{n=1}^{\infty} \frac{n+2}{n^3 - 2n + 1}$. Which of the following statements correctly uses the Limit Comparison Test to determine the convergence or divergence of this series?

(A)Comparing with $\sum_{n=1}^{\infty} \frac{1}{n}$, the limit of the ratio of terms is 1, so the series diverges.(B)Comparing with $\sum_{n=1}^{\infty} \frac{1}{n^2}$, the limit of the ratio of terms is 1, so the series converges.(C)Comparing with $\sum_{n=1}^{\infty} \frac{1}{n^2}$, the limit of the ratio of terms is 0, so the series converges.(D)Comparing with $\sum_{n=1}^{\infty} \frac{1}{n^3}$, the limit of the ratio of terms is 1, so the series converges.

34. A student attempts to determine the convergence of the series $\sum_{n=2}^{\infty} \frac{1}{n^2 - 1}$ using the Direct Comparison Test with the convergent p-series $\sum_{n=2}^{\infty} \frac{1}{n^2}$. Which of the following describes why this specific application of the Direct Comparison Test does not provide a conclusion?

(A)Because $\frac{1}{n^2 - 1} < \frac{1}{n^2}$, and establishing that a series is smaller than a convergent series confirms convergence.(B)Because $\frac{1}{n^2 - 1} > \frac{1}{n^2}$, and establishing that a series is larger than a convergent series provides no information.(C)Because the limit of the ratio $\frac{1/(n^2-1)}{1/n^2}$ is 1, which implies the test is inconclusive.(D)Because the terms of the series $\frac{1}{n^2 - 1}$ are not positive for all $n \ge 2$.

35. [Skill: 3.D | Topic: 10.7] Which of the following best describes the behavior of the infinite series $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sqrt{n}}$ ?

(A)The series converges by the Alternating Series Test.(B)The series diverges because $\frac{1}{\sqrt{n}}$ is not decreasing for all $n$.(C)The series diverges because $\lim_{n \to \infty} \frac{1}{\sqrt{n}} \neq 0$.(D)The series diverges because it is a p-series with $p = \frac{1}{2} < 1$.

36. [Skill: 3.B | Topic: 10.7] Consider the series $\sum_{n=1}^{\infty} (-1)^{n+1} a_n$, where $a_n = \frac{3n}{4n+5}$. Which of the following statements is true regarding the convergence of this series?

(A)The series converges because the terms alternate in sign and $a_n > 0$.(B)The series converges because $\lim_{n \to \infty} a_n = \frac{3}{4} < 1$.(C)The series diverges because $\lim_{n \to \infty} a_n \neq 0$.(D)The series diverges because $a_{n+1} > a_n$ for all $n$.

37. [Skill: 3.D | Topic: 10.7] Which of the following series converges by the Alternating Series Test?

(A)$\sum_{n=1}^{\infty} (-1)^n \left( \frac{n}{n+1} \right)$(B)$\sum_{n=1}^{\infty} (-1)^n \left( \frac{1}{\ln(n+1)} \right)$(C)$\sum_{n=1}^{\infty} \frac{(-1)^{2n}}{n}$(D)$\sum_{n=1}^{\infty} (-1)^n \left( 1 + \frac{1}{n} \right)$

38. [Skill: 1.C | Topic: 10.8] Consider the infinite series $\sum_{n=1}^{\infty} \frac{3^n}{(n+1)!}$. Which of the following correctly describes the result of applying the Ratio Test to this series?

(A)The limit of the ratio is 0, so the series converges.(B)The limit of the ratio is 3, so the series diverges.(C)The limit of the ratio is $\frac{1}{3}$, so the series converges.(D)The limit of the ratio is $\infty$, so the series diverges.

39. [Skill: 3.D | Topic: 10.8] Which of the following statements is true regarding the convergence of the series $\sum_{n=1}^{\infty} \frac{n^{10}}{2^n}$ ?

(A)The series diverges by the Ratio Test because $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 2 > 1$.(B)The series converges by the Ratio Test because $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \frac{1}{2} < 1$.(C)The series diverges by the Ratio Test because $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \infty$.(D)The Ratio Test is inconclusive because $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 1$.

40. [Skill: 2.C | Topic: 10.8] For which of the following series is the Ratio Test **inconclusive**?

(A)$\sum_{n=1}^{\infty} \frac{100^n}{n!}$(B)$\sum_{n=1}^{\infty} e^{-n}$(C)$\sum_{n=1}^{\infty} \frac{n}{n^3 + 1}$(D)$\sum_{n=1}^{\infty} \frac{n!}{2^n}$

41. Which of the following best describes the behavior of the series $\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{\sqrt{n}}$ ?

(A)The series converges absolutely.(B)The series converges conditionally.(C)The series diverges because the limit of the terms is not zero.(D)The series diverges because the corresponding series of absolute values diverges.

42. For which of the following values of $p$ does the series $\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^n}{n^p}$ converge conditionally?

(A)p = 0(B)p = 0.5(C)p = 1.5(D)p = 2

43. Let $\displaystyle \sum_{n=1}^{\infty} a_n$ be an infinite series. Which of the following statements must be true?

(A)If $\sum_{n=1}^{\infty} $a_n converges, then $\sum_{n=1}^{\infty} $|a_n| converges.(B)If $\sum_{n=1}^{\infty} $|a_n| converges, then $\sum_{n=1}^{\infty} $a_n converges.(C)If $\sum_{n=1}^{\infty} $|a_n| diverges, then $\sum_{n=1}^{\infty} $a_n diverges.(D)If $\lim_{n \to \infty} $a_n = 0, then $\sum_{n=1}^{\infty} $a_n converges.

44. Let $f$ be a function having derivatives of all orders for all real numbers. Let $P_3(x)$ be the third-degree Taylor polynomial for $f$ about $x=0$. The graph of $|f^{(4)}(x)|$, the absolute value of the fourth derivative of $f$, is shown above. The graph is strictly increasing on the interval $[0, 1]$ and passes through the points $(0, 2)$, $(0.5, 4)$, and $(1, 6)$. Which of the following is the Lagrange error bound for the approximation of $f(0.5)$ using $P_3(0.5)$?

(A)$\frac{1}{192}$(B)$\frac{1}{96}$(C)$\frac{1}{64}$(D)$\frac{1}{12}$

45. The infinite series $\sum_{n=1}^{infty} (-1)^{n+1} \frac{1}{n^2 + 5}$ converges to the sum $S$. Using the Alternating Series Error Bound, what is the least number of terms, $k$, needed to guarantee that the partial sum $S_k$ is within $0.01$ of $S$?

(A)8(B)9(C)10(D)11

B. Short Answer Questions (3 Questions)

Answer all parts of each question. Answers must be in essay form. Outlines or lists alone are not acceptable.

Question 46:

Question 47:

Question 48:

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