## Calculus AB and Calculus BC

## CHAPTER 10 Sequences and Series

### Practice Exercises

**Part** A. **Directions:** Answer these questions *without* using your calculator.

*Note: No* questions on sequences will appear on the BC examination. We have nevertheless chosen to include the topic in Questions 1–5 because a series and its convergence are defined in terms of sequences. Review of sequences will enhance understanding of series.

**1.** Which sequence converges?

**(A)**

**(B)**

**(C)**

**(D)**

**(E)**

**2.**

**(A)** *s _{n}* diverges by oscillation

**(B)** *s _{n}* converges to zero

**(C)**

**(D)** *s _{n}* diverges to infinity

**(E)** None of the above is true.

**3.** The sequence

**(A)** is unbounded

**(B)** is monotonic

**(C)** converges to a number less than 1

**(D)** is bounded

**(E)** diverges to infinity

**4.** Which of the following sequences diverges?

**(A)**

**(B)**

**(C)**

**(D)**

**(E)**

**5.** The sequence {*r ^{n}* } converges if and only if

**(A)** |*r*| < 1

**(B)** |*r*| 1

**(C)** −1 < *r* 1

**(D)** 0 < *r* < 1

**(E)** |*r*| > 1

**6.** is a series of constants for which Which of the following statements is always true?

**(A)** converges to a finite sum.

**(B)** equals zero.

**(C)** does not diverge to infinity.

**(D)** is a positive series.

**(E)** none of these

**7.** Note that equals

**(A)** 0

**(B)** 1

**(C)**

**(D)**

**(E)** ∞

**8.** The sum of the geometric series

**(A)**

**(B)**

**(C)** 1

**(D)**

**(E)**

**9.** Which of the following statements about series is true?

**(A)** If converges.

**(B)** If diverges.

**(C)** If diverges, then

**(D)** converges if and only if

**(E)** none of these

**10.** Which of the following series diverges?

**(A)**

**(B)**

**(C)**

**(D)**

**(E)** none of these

**11.** Which of the following series diverges?

**(A)**

**(B)**

**(C)**

**(D)** 1−1.1 + 1.21−1.331 + ···

**(E)**

**12.** Let then *S* equals

**(A)** 1

**(B)**

**(C)**

**(D)** 2

**(E)** 3

**13.** Which of the following expansions is impossible?

**(A)** in powers of *x*

**(B)** in powers of *x*

**(C)** ln *x* in powers of (*x* − 1)

**(D)** tan *x* in powers of

**(E)** ln (1 − *x*) in powers of *x*

**14.** The series converges if and only if

**(A)** *x* = 0

**(B)** 2 < *x* < 4

**(C)** *x* = 3

**(D)** 2 *x* 4

**(E)** *x* < 2 or *x* > 4

**15.** Let The radius of convergence of is

**(A)** 0

**(B)** 1

**(C)** 2

**(D)** ∞

**(E)** none of these

**16.** The coefficient of *x*^{4} in the Maclaurin series for *f* (*x*) = *e*^{−}^{x}^{/2} is

**(A)**

**(B)**

**(C)**

**(D)**

**(E)**

**17.** If an appropriate series is used to evaluate then, correct to three decimal places, the definite integral equals

**(A)** 0.009

**(B)** 0.082

**(C)** 0.098

**(D)** 0.008

**(E)** 0.090

**18.** If the series tan^{−1} is used to approximate with an error less than 0.001, then the smallest number of terms needed is

**(A)** 100

**(B)** 200

**(C)** 300

**(D)** 400

**(E)** 500

**19.** Let *f* be the Taylor polynomial *P*_{7} (*x*) of order 7 for tan^{−1} *x* about *x* = 0. Then it follows that, if −0.5 < *x* < 0.5,

**(A)** *f* (*x*) = tan^{−1} *x*

**(B)** *f* (*x*) *≤* tan^{−1} *x*

**(C)** *f* (*x*) *≥* tan^{−1} *x*

**(D)** *f* (*x*) > tan^{−1} *x* if *x* < 0 but < tan^{−1} *x* if *x* > 0

**(E)** *f* (*x*) < tan^{−1} *x* if *x* < 0 but > tan^{−1} *x* if *x* > 0

**20.** Replace the first sentence in Question 19 by “Let *f* be the Taylor polynomial *P*_{9} (*x*) of order 9 for tan^{−1} *x* about *x* = 0.” Which choice given in Question 19 is now the correct one?

**Part B. Directions:** Some of the following questions require the use of a graphing calculator.

**21.** Which of the following statements about series is false?

**(A)** where *m* is any positive integer.

**(B)** If converges, so does if *c* ≠ 0.

**(C)** If and converge, so does where *c* ≠ 0.

**(D)** If 1000 terms are added to a convergent series, the new series also converges.

**(E)** Rearranging the terms of a positive convergent series will not affect its convergence or its sum.

**22.** Which of the following series converges?

**(A)**

**(B)**

**(C)**

**(D)**

**(E)**

**23.** Which of the following series diverges?

**(A)**

**(B)**

**(C)**

**(D)**

**(E)**

**24.** For which of the following series does the Ratio Test fail?

**(A)**

**(B)**

**(C)**

**(D)**

**(E)**

**25.** Which of the following alternating series diverges?

**(A)**

**(B)**

**(C)**

**(D)**

**(E)**

**26.** Which of the following statements is true?

**(A)** If converges, then so does the series

**(B)** If a series is truncated after the *n*th term, then the error is less than the first term omitted.

**(C)** If the terms of an alternating series decrease, then the series converges.

**(D)** If *r* < 1, then the series converges.

**(E)** none of these

**27.** The power series converges if and only if

**(A)** −1 < *x* < 1

**(B)** −1 *x* 1

**(C)** −1 *x* < 1

**(D)** −1 < *x* 1

**(E)** *x* = 0

**28.** The power series

diverges

**(A)** for no real *x*

**(B)** if −2<*x* 0

**(C)** if *x* < −2 or *x* > 0

**(D)** if −2 *x* < 0

**(E)** if *x* ≠ −1

**29.** The series obtained by differentiating term by term the series

converges for

**(A)** 1 *x* 3

**(B)** 1 *x* < 3

**(C)** 1 < *x* 3

**(D)** 0 *x* 4

**(E)** none of these

**30.** The Taylor polynomial of order 3 at *x* = 0 for is

**(A)**

**(B)**

**(C)**

**(D)**

**(E)**

**31.** The Taylor polynomial of order 3 at *x* = 1 for *e ^{x}* is

**(A)**

**(B)**

**(C)**

**(D)**

**(E)**

**32.** The coefficient of in the Taylor series about of *f* (*x*) = cos *x* is

**(A)**

**(B)**

**(C)**

**(D)**

**(E)**

**33.** Which of the following series can be used to compute ln 0.8?

**(A)** ln (*x* − 1) expanded about *x* = 0

**(B)** ln *x* about *x* = 0

**(C)** ln *x* expanded about *x* = 1

**(D)** ln (*x* − 1) expanded about *x* = 1

**(E)** none of these

**34.** If *e*^{−0.1} is computed using a Maclaurin series, then, correct to three decimal places, it equals

**(A)** 0.905

**(B)** 0.950

**(C)** 0.904

**(D)** 0.900

**(E)** 0.949

**35.** The coefficient of *x*^{2} in the Maclaurin series for *e*^{sin} * ^{x}* is

**(A)** 0

**(B)** 1

**(C)**

**(D)** −1

**(E)**

**36.** Let Suppose both series converge for |*x*| < *R*. Let *x*_{0} be a number such that |*x*_{0} | < *R*. Which of the following statements is false?

**(A)** converges to *f* (*x*_{0}) + *g*(*x*_{0}).

**(B)** converges to *f* (*x*_{0})*g*(*x*_{0}).

**(C)** is continuous at *x* = *x*_{0}.

**(D)** converges to *f* *′*(*x*_{0}).

**(E)** none of these

**37.** The coefficient of (*x* − 1)^{5} in the Taylor series for *x* ln *x* about *x* = 1 is

**(A)**

**(B)**

**(C)**

**(D)**

**(E)**

**38.** The radius of convergence of the series

**(A)** 0

**(B)** 2

**(C)**

**(D)**

**(E)** ∞

**39.** If the approximate formula sin *x* = is used and |*x*| < 1 (radian), then the error is numerically less than

**(A)** 0.001

**(B)** 0.003

**(C)** 0.005

**(D)** 0.008

**(E)** 0.009

**40.** If a suitable series is used, then correct to three decimal places, is

**(A)** −0.200

**(B)** 0.180

**(C)** 0.190

**(D)** −0.190

**(E)** −0.990

**41.** The function and *f* *′*(*x*) = −*f* (*x*) for all *x*. If *f* (0) = 1, then *f* (0.2), correct to three decimal places, is

**(A)** 0.905

**(B)** 1.221

**(C)** 0.819

**(D)** 0.820

**(E)** 1.220

**42.** The sum of the series is equal to

**(A)** 0

**(B)** 1

**(C)**

**(D)**

**(E)** none of these

**43.** When is approximated by the sum of its first 300 terms, the error is closest to

**(A)** 0.001

**(B)** 0.002

**(C)** 0.005

**(D)** 0.01

**(E)** 0.02

**44.** The Taylor polynomial of order 3 at *x* = 0 for (1 + *x*)* ^{p}*, where

*p*is a constant, is

**(A)** 1 + *px* + *p*(*p* − 1)*x*^{2} + *p*(*p* − 1)(*p* − 2)*x*^{3}

**(B)**

**(C)**

**(D)**

**(E)** none of these

**45.** The Taylor series for ln (1 + 2*x*) about *x* = 0 is

**(A)**

**(B)** 2*x* − 2*x*^{2} + 8*x*^{3} − 16*x*^{4} + · · ·

**(C)** 2*x* − 4*x*^{2} + 16*x*^{3} + · · ·

**(D)**

**(E)**

**46.** The set of all values of *x* for which converges is

**(A)** only *x* = 0

**(B)** |*x*| = 2

**(C)** −2 < *x* < 2

**(D)** |*x*| > 2

**(E)** none of these

**47.** The third-order Taylor polynomial *P*_{3} (*x*) for sin *x* about is

**(A)**

**(B)**

**(C)**

**(D)**

**(E)**

**48.** Let *h* be a function for which all derivatives exist at *x* = 1. If *h*(1) = *h′* (1) = *h″* (1) = *h*′″ (1) = 6, which third-degree polynomial best approximates *h* there?

**(A)** 6 + 6*x* + 6*x*^{2} + 6*x*^{3}

**(B)** 6 + 6(*x* − 1) + 6(*x* − 1)^{2} + 6(*x* − 1)^{3}

**(C)** 6 + 6*x* + 3*x*^{2} + *x*^{3}

**(D)** 6 + 6(*x* − 1) + 3(*x* − 1)^{2} + (*x* − 1)^{3}

**(E)**