Sequences and Series: Master the Fundamentals of Mathematical Sequences and Series

Question 1

Which sequence converges?

Accepted Answer

a_n = 1/n

Answer

a_n = n^2 + 1

Answer

a_n = (-1)^n

Answer

a_n = e^n

Question 2

Which property is NOT true for convergent sequences?

Accepted Answer

Every convergent sequence is monotonic.

Answer

Every Cauchy sequence is convergent.

Answer

Every bounded sequence has a convergent subsequence.

Answer

Every subsequence of a convergent sequence is convergent.

Question 3

Which of the following series is convergent?

Accepted Answer

∑(n=1 to ∞) 1/n^2

Answer

∑(n=1 to ∞) 1/n

Answer

∑(n=1 to ∞) n

Answer

∑(n=1 to ∞) 2^n

Question 4

Find the sum of the first 10 terms of the geometric series 2 + 4 + 8 + ...

Accepted Answer

1023

Answer

1022

Answer

1024

Answer

1025

Question 5

Find the general term of the arithmetic sequence with first term 5 and common difference 3.

Accepted Answer

a_n = 5 + 3n

Answer

a_n = 5 - 3n

Answer

a_n = 5 + 2n

Answer

a_n = 5 - 2n

Question 6

State a necessary condition for a series to converge.

Accepted Answer

The terms of the series must approach 0 as n approaches infinity.

Answer

The terms of the series must be positive.

Answer

The terms of the series must alternate in sign.

Answer

The terms of the series must be decreasing.

Question 7

Find the nth term of the arithmetic sequence: 2, 5, 8, 11, ...

Accepted Answer

a_n = 2 + 3(n-1)

Answer

a_n = 5 + 3(n-1)

Answer

a_n = 8 + 3(n-1)

Answer

a_n = 11 + 3(n-1)

Question 8

Determine whether the series 1 + 2 + 4 + 8 + ... is convergent or divergent.

Accepted Answer

Divergent

Answer

Convergent

Answer

Cannot be determined

Question 9

Which of the following is the nth term of the harmonic sequence: 1, 1/2, 1/3, 1/4, ...?

Accepted Answer

1/(n+1)

Answer

1/n

Answer

n/(n+1)

Answer

n

Question 10

Which of the following is a test for convergence of alternating series?

Accepted Answer

Alternating Series Test

Answer

Ratio Test

Answer

Comparison Test

Answer

Integral Test

Question 11

Which of the following series converges? Consider (nin mathbb{N}).

Accepted Answer

Σ (1/n^2)

Answer

Σ (n!)

Answer

Σ (sin(n))

Answer

Σ (1/n)

Question 12

Find the limit of the sequence {n/(n+2)} as n approaches infinity.

Accepted Answer

1

Answer

0

Answer

Infinity

Answer

2

Question 13

Determine whether the following series is convergent or divergent: Σ (1/3^n)

Accepted Answer

Convergent

Answer

Divergent

Answer

Cannot be determined

Question 14

Which of the following is a Cauchy sequence?

Accepted Answer

{1, 1/2, 1/4, 1/8, ...}

Answer

{1, -1, 1, -1, ...}

Answer

{2, 2, 2, 2, ...}

Answer

{n, n+1, n, n+1, ...}

Question 15

Which of the following limit tests can be used to determine the convergence or divergence of the series Σ (1/n)?

Accepted Answer

Limit Comparison Test

Answer

Comparison Test

Answer

Ratio Test

Answer

Squeeze Theorem

Question 16

Which test is specifically designed to determine if a sequence diverges?

Accepted Answer

Divergence test

Answer

Cauchy criterion

Answer

Ratio test

Answer

Root test

Question 17

Determine whether the series 1 + 1/2 + 1/4 + 1/8 + ... converges or diverges.

Accepted Answer

Converges to 1

Answer

Converges to 2

Answer

Diverges to infinity

Answer

Converges to 0

Question 18

Determine if the series 1 + 2 + 4 + 8 + ... is convergent or divergent.

Accepted Answer

Diverges to infinity

Answer

Converges to 15

Answer

Converges to 31

Answer

Converges to 63

Question 19

Find the interval of convergence for the power series Σ (x - 3)^n / n.

Accepted Answer

(-2, 4]

Answer

(-2, 4)

Answer

[2, 4)

Answer

[2, 4]

Question 20

Which of the following is a valid method to test for convergence of an alternating series?

Accepted Answer

Alternating Series Test

Answer

Ratio Test

Answer

Root Test

Answer

Integral Test

Question 21

Determine the sum of the first 10 terms of the series: 1 + 1/2 + 1/4 + 1/8 + ...

Accepted Answer

1023/512

Answer

512/1023

Answer

1024/511

Answer

511/1024

Question 22

Identify the limit of the sequence: 1, 1/2, 1/4, 1/8, ...

Accepted Answer

0

Answer

1

Answer

1/2

Answer

Infinity

Question 23

Which of the following sequences is not bounded?

Accepted Answer

2, 4, 8, 16, ...

Answer

1, 2, 3, 4, ...

Answer

-1, -2, -3, -4, ...

Answer

1/2, 1/4, 1/8, 1/16, ...

Question 24

Classify the type of series: 1 + 2 + 4 + 8 + ...

Accepted Answer

Geometric

Answer

Arithmetic

Answer

Harmonic

Answer

Alternating

Question 25

Evaluate the following limit: lim (n->∞) (1 + 1/n)^n

Accepted Answer

e

Answer

0

Answer

1

Answer

Infinity

Question 26

Determine whether the following series is convergent or divergent: 1 + 1/2 + 1/4 + 1/8 + ...

Accepted Answer

Converges

Answer

Diverges

Answer

Cannot be determined

Question 27

Write the Maclaurin series expansion for the function e^x.

Accepted Answer

Σ x^n / n!

Answer

Σ (-1)^n x^n / n

Answer

Σ n x^n / n!

Answer

Σ (-1)^n n x^n / n!

Question 28

What is the 10th term (a_10) of the arithmetic sequence 2, 5, 8, 11, ...?

Accepted Answer

29

Answer

26

Answer

32

Question 29

Find the sum of the first 20 terms of the geometric series 1 + 2 + 4 + 8 + ...

Accepted Answer

1,048,575

Answer

2,097,151

Answer

1,023

Answer

524,287

Question 30

Which of the following sequences converges?

Accepted Answer

1/2, 1/4, 1/8, 1/16, ...

Answer

1, 1/2, 1/3, 1/4, ...

Answer

1, -1, 1, -1, ...

Answer

1, 2, 4, 8, ...

Question 31

A ball is dropped from a height of 10 meters. Each time it bounces, it rebounds to 3/4 of its previous height. What is the total distance the ball travels before coming to rest?

Accepted Answer

40 meters

Answer

20 meters

Answer

10 meters

Answer

30 meters

Question 32

What is the general term (nth term) of the sequence 1, 4, 9, 16, ...?

Accepted Answer

n^2

Answer

2n + 1

Answer

n^3

Answer

n + 3

Question 33

Find the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + ...

Accepted Answer

3/2

Answer

1/2

Answer

3

Answer

2

Question 34

A sequence is defined by the recursive formula a_1 = 2 and a_n = a_(n-1) + 3. What is the value of a_5?

Accepted Answer

14

Answer

11

Answer

17

Answer

20

Question 35

The first term of an arithmetic sequence is 5 and the common difference is 4. What is the sum of the first 10 terms of this sequence?

Accepted Answer

245

Answer

125

Answer

195

Question 36

Which of the following statements is true about the sequence 1, 1/2, 1/4, 1/8, ...?

Accepted Answer

It is a geometric sequence with a common ratio of 1/2.

Answer

It is neither an arithmetic nor a geometric sequence.

Answer

It is an arithmetic sequence with a common difference of 1/2.

Question 37

What is the 10th term of the arithmetic sequence 2, 5, 8, 11, ...?

Accepted Answer

29

Answer

31

Answer

28

Answer

30

Question 38

What is the sum of the first 5 terms of the geometric sequence 3, 6, 12, 24, ...?

Accepted Answer

93

Answer

96

Answer

90

Answer

87

Question 39

An infinite geometric series has a sum of 10 and a first term of 5. What is the common ratio?

Accepted Answer

1/2

Answer

1/3

Answer

2/3

Answer

1/4

Question 40

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

Accepted Answer

2

Answer

3

Answer

1

Answer

1.5

Question 41

The sum of the first 'n' terms of an arithmetic series is given by Sn = 3n^2 + 5n. What is the common difference of the series?

Accepted Answer

6

Answer

3

Answer

5

Answer

7

Question 42

Which of the following sequences is convergent?

Accepted Answer

1/2, 1/4, 1/8, 1/16, ...

Answer

1, -1, 1, -1, ...

Answer

2, 4, 8, 16, ...

Answer

1, 2, 3, 4, ...

Question 43

The sequence {an} is defined by an = (n + 1)/n. What is the limit of the sequence as n approaches infinity?

Accepted Answer

1

Answer

0

Answer

2

Answer

∞

Question 44

A company's annual profit increases by 10% each year. If the initial profit was $100,000, what will the total profit be after 5 years?

Accepted Answer

Approximately $161,051

Answer

Approximately $150,000

Answer

Approximately $175,000

Answer

Approximately $200,000

Question 45

A ball is dropped from a height of 10 meters. Each time it bounces, it reaches a height that is 3/4 of the previous height. What is the total distance the ball travels before coming to rest?

Accepted Answer

40 meters

Answer

60 meters

Answer

30 meters

Answer

50 meters

Question 46

Which of the following represents a geometric sequence?

Accepted Answer

a, ar, ar², ar³, ...

Answer

a, a², a³, a⁴, ...

Answer

a, a + d, a + 2d, a + 3d, ...

Question 47

Find the nth term of an arithmetic sequence with first term a and common difference d.

Accepted Answer

a + (n - 1)d

Answer

a - (n - 1)d

Answer

a + (n + 1)d

Answer

a - (n + 1)d

Question 48

Determine whether the sequence defined by a_n = 2n - 5 is convergent or divergent.

Accepted Answer

Divergent

Answer

Convergent to 2

Answer

Convergent to 0

Answer

Convergent to -5

Question 49

The sum of the first n terms of a geometric series with first term a and common ratio r is given by:

Accepted Answer

a(1 - r^n) / (1 - r)

Answer

a(1 + r^n) / (1 + r)

Answer

a + r + r² + ... + r^n

Answer

(a - r)n / (1 - r)

Question 50

Which of the following series is convergent?

Accepted Answer

∑ (1/n^2)

Answer

∑ (1/n)

Answer

∑ (n)

Answer

∑ (n^2)

Question 51

Find the interval of convergence of the power series ∑ (x - 2)^n / n^2.

Accepted Answer

(1, 3)

Answer

(-2, 2)

Answer

(-1, 1)

Answer

(0, 4)

Question 52

Which of the following limits is used to represent the sum of an infinite series?

Accepted Answer

lim (n→∞) Sn

Answer

lim (n→∞) ∫[a_0, a_n] f(x) dx

Answer

lim (n→∞) (a_n + a_(n+1))

Answer

lim (n→∞) a_n

Question 53

The Taylor series expansion of f(x) = e^x around x = 0 is:

Accepted Answer

∑ (x^n / n!)

Answer

∑ (x^n)

Answer

∑ (-x^n / n!)

Answer

∑ (x^n / (n + 1)!)

Question 54

Determine whether the sequence a_n = sin(n) is Cauchy.

Accepted Answer

Not Cauchy

Answer

Uniformly convergent

Answer

Absolutely convergent

Answer

Cauchy