Sequences and Series: Master the Concepts with Interactive Quizzes

Question 1

Which of the following best defines a convergent sequence?

Accepted Answer

A sequence whose terms approach a fixed value as the term number increases indefinitely

Answer

A sequence whose terms oscillate between two fixed values as the term number increases indefinitely

Answer

A sequence whose terms increase without bound as the term number increases indefinitely

Answer

A sequence whose terms decrease without bound as the term number increases indefinitely

Question 2

Which of the following is NOT a commonly used test for sequence convergence?

Accepted Answer

Term divergence test

Answer

Cauchy criterion

Answer

Ratio test

Answer

Root test

Question 3

What is the sum of the convergent geometric series ∑_{n=1}^∞ (-1)^n / 2^n?

Accepted Answer

1/2

Answer

1

Answer

-1

Answer

-1/2

Question 4

What is the interval of convergence for the power series ∑_{n=1}^∞ x^n / n?

Accepted Answer

(-1, 1)

Answer

[-1, 1]

Answer

[-∞, ∞]

Answer

(-∞, ∞)

Question 5

Which of the following functions is represented by the power series ∑_{n=0}^∞ x^n / n!

Accepted Answer

e^x

Answer

sin(x)

Answer

cos(x)

Answer

ln(x)

Question 6

Determine if the sequence defined by an = (-1)^(n+1) converges or diverges:

Accepted Answer

Diverges

Answer

Converges to -1

Answer

Converges to 1

Answer

Converges to 0

Question 7

Find the sum of the first 10 terms of the geometric series 2, 6, 18, 54, ...

Accepted Answer

1023

Answer

1022

Answer

1024

Answer

1025

Question 8

Which of the following is a telescoping series?

Accepted Answer

∑ (1/n - 1/(n+1))

Answer

∑ (n^2 - (n-1)^2)

Answer

∑ (n/(n+1))

Answer

∑ (n^2 + 1)

Question 9

Which series is conditionally convergent?

Accepted Answer

∑ (-1)^n / n

Answer

∑ (-1)^n

Answer

∑ 1/n^2

Answer

∑ 1/n

Question 10

Which sequence has a subsequence that converges to 0?

Accepted Answer

an = 1/n

Answer

an = sin(n)

Answer

an = n

Answer

an = e^n

Question 11

Determine the limit of the sequence {n^2 + 1} as n approaches infinity:

Accepted Answer

Infinity

Answer

-1

Answer

0

Answer

Does not exist

Question 12

A sufficient condition for a series to be convergent is:

Accepted Answer

Absolute convergence

Answer

Conditional convergence

Answer

Ratio Test

Answer

Root Test

Question 13

Which of the following series is a geometric series?

Accepted Answer

1 + (1/2) + (1/4) + ...

Answer

1 + (1/2) + (1/3) + ...

Answer

1 - (1/2) + (1/4) - ...

Answer

1 - (1/3) + (1/5) - ...

Question 14

The Limit Comparison Test can be used to compare two series that have:

Accepted Answer

Positive terms

Answer

Negative terms

Answer

Alternating terms

Answer

Bounded terms

Question 15

The Ratio Test is applicable to:

Accepted Answer

Geometric series

Answer

Harmonic series

Answer

Telescoping series

Answer

p-series

Question 16

State the Alternating Series Test.

Accepted Answer

If the terms of the series are alternately positive and negative and decrease in absolute value, then the series converges.

Answer

If the terms of the series are alternately positive and negative and increase in absolute value, then the series converges.

Answer

If the terms of the series are all positive and decrease in absolute value, then the series converges.

Question 17

Determine whether the series Σ(cos(nπ / 2)) converges or diverges.

Accepted Answer

Diverges

Answer

Converges

Question 18

Identify a necessary condition for a sequence to converge.

Accepted Answer

The limit of the sequence exists.

Answer

The sequence is bounded.

Answer

The sequence is monotonic.

Question 19

Determine whether the series ∑(n=1 to ∞) (1/n^2) is convergent or divergent.

Accepted Answer

Convergent

Answer

Divergent

Answer

Cannot be determined

Question 20

Which of the following sequences is arithmetic?

Accepted Answer

{-2, -1, 0, 1, ...}

Answer

{n^2}

Answer

{n / (n+1)}

Answer

{(-1)^n}

Question 21

Identify the common ratio of the geometric series 2 + 6 + 18 + ...

Accepted Answer

3

Answer

2

Answer

6

Answer

9

Question 22

Determine if the series ∑(n=1 to ∞) (n^2)/(2^n) is convergent or divergent using the Ratio Test.

Accepted Answer

Convergent

Answer

Divergent

Answer

Cannot be determined

Question 23

Determine the convergence or divergence of the alternating series ∑(n=1 to ∞) (-1)^n / n

Accepted Answer

Convergent

Answer

Divergent

Answer

Cannot be determined

Question 24

Identify a telescoping series from the following:

Accepted Answer

∑(n=1 to ∞) n/(n+1)

Answer

∑(n=1 to ∞) n

Answer

∑(n=1 to ∞) 1/n

Answer

∑(n=1 to ∞) 1/n^2

Question 25

The sequence {(-1)^n / n} alternates between -1 and 1. As n gets larger, the denominator grows, making the fraction closer to zero. Thus, the limit of the sequence is:

Accepted Answer

0

Answer

1

Answer

-1

Answer

Does not exist

Question 26

A sequence is defined as {n^2 - 2n + 1, n^2 + 2n + 1, n^2 - 2n + 1, n^2 + 2n + 1, ...}. Since the difference between consecutive terms is constant (4), the sequence is classified as:

Accepted Answer

Arithmetic

Answer

Geometric

Answer

Alternating

Answer

Oscillating

Question 27

The nth term of an arithmetic sequence is given by the formula a_n = a_1 + (n-1)d, where a_1 is the first term and d is the common difference. The formula for the nth term is:

Accepted Answer

a_n = a_1 + (n-1)d

Answer

a_n = a_1 + nd

Answer

a_n = a_1 - (n-1)d

Answer

a_n = a_1 - nd

Question 28

Find the sum of the first 10 terms of the geometric series 3 + 6 + 12 + ...

Accepted Answer

3072

Answer

1536

Answer

768

Answer

384

Question 29

Find the derivative of the sequence {aₙ} = (3n² + 2) / (n³ - 1)

Accepted Answer

(3n² - 9n - 1) / (n³ - 1)²

Answer

(6n) / (n³ - 1)

Answer

(9n) / (n³ - 1)

Answer

(3n² + 2) / (n³ - 1)²

Question 30

Which of the following is NOT a necessary condition for a series to converge?

Accepted Answer

The sum of the first n terms is less than n

Answer

The sequence of partial sums is bounded

Answer

The sequence of partial sums approaches a finite limit

Answer

The terms of the series approach zero as n → ∞

Question 31

Which of the following series represents the Taylor series expansion of sin(x) about x = 0?

Accepted Answer

∑ (-1)ⁿ (xⁿ) / (n!)

Answer

∑ (xⁿ) / n!

Answer

∑ (x²ⁿ) / (2n)!

Answer

∑ (-1)ⁿ (xⁿ+¹) / (n+1)!

Question 32

Use the Ratio Test to determine if the series ∑(n=1 to ∞) (n^2 + 1) / (n^3 + 2n) converges or diverges.

Accepted Answer

Converges

Answer

Diverges

Answer

Cannot be determined

Question 33

Find the limit of the alternating harmonic series 1 - 1/2 + 1/3 - 1/4 + ... as n approaches infinity

Accepted Answer

ln(2)

Answer

0

Answer

1

Answer

Does not exist

Question 34

Which of the following sequences is Cauchy?

Accepted Answer

{1/n}

Answer

{(-1)^n}

Answer

{n sin(n)}

Answer

{n^2 + 3n}

Question 35

Use the appropriate test to determine the convergence or divergence of the series ∑_n=1^∞ (1/√n).

Accepted Answer

Divergent by the Integral Test

Answer

Convergent by the Integral Test

Answer

Convergent by the Comparison Test

Answer

Divergent by the Comparison Test

Question 36

Determine whether the series ∑_n=1^∞ (1/(n(n+1))) is convergent.

Accepted Answer

Converges to 0

Answer

Converges to 1

Answer

Diverges

Answer

Converges to a value that cannot be simplified

Question 37

Which test can be used to determine the convergence of the alternating series ∑_n=1^∞ (-1)^n-1 / n?

Accepted Answer

Alternating Series Test

Answer

Ratio Test

Answer

Root Test

Answer

Integral Test

Question 38

Identify the INCORRECT statement about the p-series.

Accepted Answer

The p-series converges if p = 1.

Answer

The p-series converges if p > 1.

Answer

The p-series diverges if p < 1.

Answer

The p-series diverges if p = 1.

Question 39

Which series is an example of a harmonic series?

Accepted Answer

∑_n=1^∞ 1/n

Answer

∑_n=1^∞ 1/n²

Answer

∑_n=1^∞ 1/2ⁿ

Answer

∑_n=1^∞ 1/n!

Question 40

Find the sum of the infinite geometric series 1 + 1/2 + 1/4 + 1/8 + ...

Accepted Answer

2

Answer

∞

Answer

1/2

Answer

1

Question 41

Which of the following sequences is an arithmetic sequence?

Accepted Answer

2, 5, 8, 11, ...

Answer

1, 4, 9, 16, ...

Answer

1, 2, 4, 8, ...

Answer

1, 1/2, 1/4, 1/8, ...

Question 42

Find the 10th term of the arithmetic sequence 3, 7, 11, 15, ...

Accepted Answer

39

Answer

47

Answer

43

Answer

35

Question 43

Which of the following series converges?

Accepted Answer

1 + 1/2 + 1/4 + 1/8 + ...

Answer

1 + 2 + 4 + 8 + ...

Answer

1 + 1/2 + 1/3 + 1/4 + ...

Answer

1 + 2 + 3 + 4 + ...

Question 44

Find the sum of the first 5 terms of the geometric series 2 + 6 + 18 + ...

Accepted Answer

242

Answer

362

Answer

122

Answer

482

Question 45

Which of the following tests can be used to determine the convergence of the series ∑(n=1 to ∞) 1/n^2?

Accepted Answer

p-series test

Answer

comparison test

Answer

ratio test

Answer

integral test

Question 46

Find the Maclaurin series expansion for e^x

Accepted Answer

∑(n=0 to ∞) x^n/n!

Answer

∑(n=0 to ∞) x^(n+1)/(n+1)!

Answer

∑(n=0 to ∞) x^(2n)/n!

Question 47

Find the limit of the sequence {(-1)^n/n} as n approaches infinity

Accepted Answer

0

Answer

1

Answer

-1

Answer

The limit does not exist

Question 48

Find the value of F(5) in the Fibonacci sequence, where F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1

Accepted Answer

5

Answer

3

Answer

8

Question 49

Does the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... converge or diverge?

Accepted Answer

Diverges

Answer

Converges to 2

Answer

Converges to ∞

Answer

Converges to 1

Question 50

What is the next term in the arithmetic sequence 2, 5, 8, 11, ...?

Accepted Answer

14

Answer

13

Answer

15

Answer

12

Question 51

Find the sum of the first 10 terms of the geometric series 3 + 6 + 12 + ...

Accepted Answer

3069

Answer

3066

Answer

1533

Answer

6135

Question 52

Which of the following series converges?

Accepted Answer

∑ (1/n^2)

Answer

∑ (1/n)

Answer

∑ (n)

Answer

∑ (2^n)

Question 53

The sequence (1/2)^n is:

Accepted Answer

Convergent

Answer

Bounded

Answer

Oscillatory

Answer

Divergent

Question 54

Determine the limit of the sequence {a_n} = (n^2 + 1)/(n^2 - 1) as n approaches infinity.

Accepted Answer

1

Answer

-1

Answer

∞

Answer

0

Question 55

What is the sum of the infinite geometric series 1/2 + 1/4 + 1/8 + ...?

Accepted Answer

1

Answer

2

Answer

∞

Answer

1/2

Question 56

Which test could be used to determine the convergence of the series ∑ (n!)/(n^n)?

Accepted Answer

Ratio Test

Answer

Comparison Test

Answer

Integral Test

Question 57

The harmonic series ∑ (1/n) is:

Accepted Answer

Divergent

Answer

Absolutely Convergent

Answer

Convergent

Answer

Conditionally Convergent

Question 58

Find the sum of the first 5 terms of the sequence defined by the recursive formula: a_1 = 2, a_n = 2a_{n-1} + 1.

Accepted Answer

47

Answer

31

Answer

63

Answer

23

Question 59

Which of the following is NOT a property of convergent sequences?

Accepted Answer

They must be bounded above and below.

Answer

They must be bounded.

Answer

They must be monotonic.

Answer

They must have a finite limit.