Calc 2 Test 3 Review

Question 1

What is a geometric series?

Answer 1

A geometric series is the sum of the terms of a geometric sequence, where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Question 2

True or False: A geometric series converges if the common ratio is greater than 1.

Answer 2

False

Question 3

Fill in the blank: A geometric series converges when the absolute value of the common ratio, r, is ______.

Answer 3

less than 1

Question 4

What is the formula for the sum of an infinite geometric series?

Answer 4

The sum S of an infinite geometric series can be calculated using the formula S = a / (1 - r), where a is the first term and r is the common ratio.

Question 5

Multiple Choice: Which of the following values of r will make the geometric series converge? A) 0.5 B) 1.5 C) 2 D) -1

Answer 5

A) 0.5

Question 6

What is a p-series?

Answer 6

A p-series is a series of the form ∑ (1/n^p) where n = 1 to infinity and p is a positive constant.

Question 7

True or False: A p-series converges if p > 1.

Answer 7

True.

Question 8

Fill in the blank: A p-series diverges if p _____ 1.

Answer 8

≤

Question 9

For which values of p does the p-series converge?

Answer 9

The p-series converges for p > 1.

Question 10

What happens to a p-series when p = 1?

Answer 10

The p-series diverges when p = 1.

Question 11

What is the divergence test used for in calculus?

Answer 11

The divergence test is used to determine whether a series diverges.

Question 12

True or False: If the limit of the terms of a series does not approach zero, the series must diverge.

Answer 12

True

Question 13

Fill in the blank: The divergence test states that if ( lim_{n \to \infty} a_n \neq 0 ) or does not exist, then the series ( \sum a_n ) __________.

Answer 13

diverges

Question 14

What is the necessary condition for the divergence test to apply?

Answer 14

The necessary condition is that the limit of the series terms must be evaluated.

Question 15

Multiple Choice: Which of the following is a conclusion of the divergence test? A) The series converges, B) The series diverges, C) The test is inconclusive.

Answer 15

B) The series diverges

Question 16

What is the integral test used for?

Answer 16

The integral test is used to determine the convergence or divergence of an infinite series.

Question 17

True or False: The integral test can only be applied to positive, continuous, and decreasing functions.

Answer 17

True

Question 18

Fill in the blank: The integral test states that if the series ( \sum_{n=1}^{\infty} a_n ) converges, then the improper integral ( \int_{1}^{\infty} f(x) \, dx ) converges, where ( f(x) = a_n ) for ( n = x ).

Answer 18

converges

Question 19

What must be true about the function ( f(x) ) for the integral test to be applicable?

Answer 19

It must be positive, continuous, and decreasing on the interval.

Question 20

Which of the following series can be analyzed using the integral test? A) ( \sum_{n=1}^{\infty} \frac{1}{n^2} ) B) ( \sum_{n=1}^{\infty} \frac{1}{n} ) C) Both A and B

Answer 20

C) Both A and B

Question 21

What is the comparison test used for in mathematics?

Answer 21

The comparison test is used to determine the convergence or divergence of infinite series.

Question 22

True or False: The comparison test can only be applied to positive series.

Answer 22

True

Question 23

Fill in the blank: The comparison test compares a given series to a _____ series.

Answer 23

benchmark

Question 24

What are the two main conditions under which the comparison test is applicable?

Answer 24

1. If the series you are testing is less than a convergent series, it converges. 2. If it is greater than a divergent series, it diverges.

Question 25

In the comparison test, if a series Σa_n converges and a_n ≤ b_n for all n, what can be concluded about Σb_n?

Answer 25

Σb_n also converges.

Question 26

What is the limit comparison test used for?

Answer 26

The limit comparison test is used to determine the convergence or divergence of an infinite series.

Question 27

True or False: The limit comparison test requires comparing two series with non-negative terms.

Answer 27

True

Question 28

Fill in the blank: If the limit of the ratio of two series as n approaches infinity is a positive finite number, then both series either converge or diverge. This is known as the ________ comparison test.

Answer 28

limit

Question 29

What is the formula used in the limit comparison test?

Answer 29

The formula is lim (n -> ∞) a_n / b_n, where a_n and b_n are the terms of the two series.

Question 30

If lim (n -> ∞) a_n / b_n = 0 and b_n converges, what can we conclude about a_n?

Answer 30

a_n converges.

Question 31

What is the ratio test used for in mathematics?

Answer 31

The ratio test is used to determine the convergence or divergence of infinite series.

Question 32

True or False: The ratio test can only be applied to series with positive terms.

Answer 32

True

Question 33

Fill in the blank: The ratio test involves calculating the limit of the absolute value of the ratio of consecutive terms, expressed as ______.

Answer 33

L = lim (n -> ∞) |a_(n+1)/a_n|

Question 34

If the limit L from the ratio test is less than 1, what can be concluded about the series?

Answer 34

The series converges.

Question 35

What happens if the limit L from the ratio test is greater than 1?

Answer 35

The series diverges.

Question 36

What is the root test used for?

Answer 36

The root test is used to determine the convergence or divergence of infinite series.

Question 37

True or False: The root test involves taking the nth root of the absolute value of the terms in a series.

Answer 37

True

Question 38

Fill in the blank: The root test states that if lim sup (n→∞) √[n]{|a_n|} < 1, then the series ∑ a_n _______.

Answer 38

converges

Question 39

What does it mean if lim sup (n→∞) √[n]{|a_n|} > 1?

Answer 39

If lim sup (n→∞) √[n]{|a_n|} > 1, then the series ∑ a_n diverges.

Question 40

In the root test, if lim sup (n→∞) √[n]{|a_n|} = 1, what can be concluded?

Answer 40

No conclusion can be drawn; the test is inconclusive.

Question 41

What is the purpose of the alternating series test?

Answer 41

To determine the convergence or divergence of an alternating series.

Question 42

True or False: An alternating series converges if the absolute values of its terms decrease to zero.

Answer 42

True.

Question 43

Fill in the blank: The alternating series test requires that the terms of the series alternate in _______.

Answer 43

sign.

Question 44

What are the two main conditions of the alternating series test for convergence?

Answer 44

The terms must decrease in absolute value and approach zero.

Question 45

Multiple Choice: Which of the following is NOT a requirement for the alternating series test? A) Terms must alternate in sign B) Terms must be positive C) Terms must decrease in absolute value D) Terms must approach zero

Answer 45

B) Terms must be positive.

Question 46

What is the general form of a convergent alternating series?

Answer 46

An alternating series is typically expressed as the sum of terms in the form (-1)^n * a_n, where a_n is a sequence of positive terms that converges to 0.

Question 47

True or False: The error bound for a convergent alternating series is given by the absolute value of the first omitted term.

Answer 47

True

Question 48

Fill in the blank: The error in approximating the sum of a convergent alternating series can be bounded by _____.

Answer 48

the absolute value of the first term not included in the sum.

Question 49

What condition must the terms a_n satisfy for the alternating series test to apply?

Answer 49

The terms a_n must be positive, decreasing, and converge to 0.

Question 50

Multiple Choice: Which of the following is used to estimate the error in a convergent alternating series? A) The last term included in the sum B) The first term not included in the sum C) The sum of all terms D) The average of all terms

Answer 50

B) The first term not included in the sum

Question 51

What is a Taylor polynomial of degree n for a function f(x) at x = a?

Answer 51

It is a polynomial approximation of the function f(x) around the point x = a, using derivatives of f at that point.

Question 52

True or False: The Taylor polynomial of degree n includes terms up to the nth derivative of f evaluated at x = a.

Answer 52

True

Question 53

Fill in the blank: The formula for the Taylor polynomial of degree n is given by P_n(x) = f(a) + f'(a)(x-a) + ________.

Answer 53

f''(a)(x-a)^2/2! + ... + f^(n)(a)(x-a)^n/n!

Question 54

What is the first term in the Taylor polynomial expansion?

Answer 54

The value of the function at the point a, which is f(a).

Question 55

Multiple Choice: Which of the following best describes the purpose of a Taylor polynomial?

Answer 55

To approximate a function near a specific point.

Question 56

What is the general form of the Maclaurin series for the function e^x?

Answer 56

The Maclaurin series for e^x is given by the formula e^x = Σ (x^n / n!) from n=0 to ∞.

Question 57

True or False: The Maclaurin series for e^x converges for all real values of x.

Answer 57

True

Question 58

Fill in the blank: The Maclaurin series for e^x starts with the term ______.

Answer 58

1

Question 59

What is the value of the second derivative of e^x at x=0?

Answer 59

The second derivative of e^x at x=0 is 1.

Question 60

Choose the correct term in the Maclaurin series for e^x: e^x = 1 + x + ____ + ...

Answer 60

x^2 / 2!

Question 61

What is the general formula for the Maclaurin series of a function f(x)?

Answer 61

The Maclaurin series of a function f(x) is given by f(x) = f(0) + f'(0)x + f''(0)x^2/2! + f'''(0)x^3/3! + ...

Question 62

True or False: The Maclaurin series for sin(x) includes only even-powered terms.

Answer 62

False

Question 63

Fill in the blank: The Maclaurin series for sin(x) is ______.

Answer 63

sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ...

Question 64

What is the value of sin(0) in the Maclaurin series expansion?

Answer 64

0

Question 65

Multiple choice: Which of the following is a term in the Maclaurin series for sin(x)? A) x^2/2! B) -x^3/3! C) x^4/4! D) x^6/6!

Answer 65

-x^3/3!

Question 66

What is the Maclaurin series for cos(x)?

Answer 66

The Maclaurin series for cos(x) is given by: cos(x) = 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + ... = Σ((-1)^n * (x^(2n)) / (2n)!, n=0 to ∞.

Question 67

True or False: The Maclaurin series for cos(x) contains only even powers of x.

Answer 67

True.

Question 68

Fill in the blank: The general term of the Maclaurin series for cos(x) can be expressed as ______.

Answer 68

(-1)^n * (x^(2n)) / (2n)!

Question 69

What is the value of cos(0) using the Maclaurin series?

Answer 69

cos(0) = 1.

Question 70

Multiple Choice: Which of the following is the correct formula for the Maclaurin series for cos(x)? A) Σ(x^n/n!), B) Σ((-1)^n * (x^(2n)) / (2n)!), C) Σ(x^n), D) Σ(x^(2n)/n!)

Answer 70

B) Σ((-1)^n * (x^(2n)) / (2n)!)

Question 71

What is the Maclaurin series for the function 1/(1-x)?

Answer 71

The Maclaurin series for 1/(1-x) is the sum of the series Σ (x^n) from n=0 to ∞.

Question 72

True or False: The Maclaurin series for 1/(1-x) converges for |x| < 1.

Answer 72

True

Question 73

Fill in the blank: The Maclaurin series for 1/(1-x) can be expressed as _____ when expanded.

Answer 73

1 + x + x^2 + x^3 + ...

Question 74

What is the radius of convergence for the Maclaurin series of 1/(1-x)?

Answer 74

The radius of convergence is 1.

Question 75

Choose the correct option: The nth term of the Maclaurin series for 1/(1-x) is represented as which of the following? a) x^n b) n! c) n^2 d) 1/n

Answer 75

a) x^n

Question 76

What is the general formula for a Taylor polynomial of degree n for a function f(x) at a point a?

Answer 76

P_n(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ... + f^(n)(a)(x-a)^n/n!

Question 77

True or False: A Taylor polynomial can only be used to approximate functions that are polynomial in nature.

Answer 77

False

Question 78

Fill in the blank: The Taylor polynomial of degree 2 for a function f(x) at a point a is also known as the _______ approximation.

Answer 78

quadratic

Question 79

What is the first step in finding a Taylor polynomial approximation of f(x) at a value a?

Answer 79

Calculate f(a) and the derivatives f'(a), f''(a), ..., f^(n)(a) at the point a.

Question 80

Multiple Choice: Which of the following is NOT a component of a Taylor polynomial?

Answer 80

f^(n+1)(a)(x-a)^(n+1)/(n+1)!

Question 81

What is the first step to find the interval of convergence for a power series?

Answer 81

Identify the power series and find its general term.

Question 82

True or False: The Ratio Test is commonly used to determine the interval of convergence.

Answer 82

True

Question 83

Fill in the blank: The interval of convergence may include endpoints, which must be tested separately for __________.

Answer 83

convergence

Question 84

What is the formula used in the Ratio Test for convergence of a power series?

Answer 84

L = lim (n -> ∞) |a_(n+1)/a_n| where L is the limit.

Question 85

If the limit L from the Ratio Test is less than 1, what can be concluded about the series?

Answer 85

The series converges absolutely.