Sequences and Series: A Comprehensive Guide for Biology Students

Question 1

Define a sequence.

Accepted Answer

An ordered set of numbers

Answer

A function that maps numbers to numbers

Answer

The limit of a function as the input approaches a specific value

Answer

The sum of the terms of a series

Question 2

Which sequence converges?

Accepted Answer

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

Answer

{1, 2, 3, 4, ...}

Answer

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

Answer

{1, 4, 9, 16, ...}

Question 3

Which series diverges?

Accepted Answer

1 + 2 + 4 + 8 + ...

Answer

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

Answer

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

Answer

1/2 + 1/4 + 1/8 + 1/16 + ...

Question 4

Calculate the sum of the first 10 terms of the geometric sequence 1, 2, 4, 8, ...

Accepted Answer

1023

Answer

511

Answer

1022

Answer

2046

Question 5

Find the limit of the sequence {n/(n+1)} as n approaches infinity.

Accepted Answer

1

Answer

0

Answer

e

Answer

∞

Question 6

Write the Taylor series expansion of f(x) = e^x.

Accepted Answer

1 + x + x^2/2! + x^3/3! + ...

Answer

x + x^2/2! + x^3/3! + x^4/4! + ...

Answer

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

Answer

x + x^2 + x^3 + x^4 + ...

Question 7

Which of the following is NOT a type of sequence?

Accepted Answer

List of prime numbers

Answer

Arithmetic sequence

Answer

Geometric sequence

Answer

Fibonacci sequence

Question 8

What is the sum of the first 10 terms of the Fibonacci sequence?

Accepted Answer

55

Answer

89

Answer

144

Answer

233

Question 9

Which of the following series is convergent?

Accepted Answer

{1/n | n ∈ ℕ}

Answer

{n² | n ∈ ℕ}

Answer

{(-1)^n | n ∈ ℕ}

Answer

{√n | n ∈ ℕ}

Question 10

What is the radius of convergence of the power series {∑(n=0 to ∞) (x - 2)^n / n!}

Accepted Answer

2

Answer

0

Answer

1

Answer

∞

Question 11

Which of the following is a telescoping series?

Accepted Answer

{∑(n=1 to ∞) 1 / n(n+1)}

Answer

{∑(n=1 to ∞) n²}

Answer

{∑(n=1 to ∞) (-1)^n}

Answer

{∑(n=1 to ∞) 2^n}

Question 12

A convergent sequence is defined as:

Accepted Answer

A sequence that approaches a fixed limit as the number of terms approaches infinity

Answer

A sequence that oscillates between two or more values

Answer

A sequence that increases or decreases without bound

Answer

A sequence that has a constant value for all terms

Question 13

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

Accepted Answer

Convergent

Answer

Divergent

Answer

Cannot be determined

Question 14

Regarding the sequence {a_n} defined as a_n = (2n-1)/(n+2), which statement is correct?

Accepted Answer

It converges to 2

Answer

It diverges to infinity

Answer

It diverges to negative infinity

Answer

It oscillates

Question 15

Determine the convergence or divergence of the geometric series ∑(n=1 to infinity) (1/2)^n:

Accepted Answer

Converges with sum 1

Answer

Converges with sum 2

Answer

Diverges

Answer

Cannot be determined

Question 16

Which of the following sequences is Cauchy?

Accepted Answer

{a_n} = 1 / n^2

Answer

{a_n} = n^2

Answer

{a_n} = (-1)^n

Answer

{a_n} = 2^n

Question 17

Determine the radius of convergence of the power series ∑(n=0 to infinity) x^n / n !:

Accepted Answer

∞

Answer

0

Answer

1

Answer

Cannot be determined

Question 18

What is the defining characteristic that distinguishes convergent sequences from divergent sequences?

Accepted Answer

Convergent sequences approach a finite value as n increases without bound, while divergent sequences do not.

Answer

Divergent sequences approach a finite value as n increases without bound, while convergent sequences do not.

Answer

Convergent sequences are bounded, while divergent sequences are not.

Answer

Divergent sequences are monotonic, while convergent sequences are not.

Question 19

Which of the following formulas represents the nth term of an arithmetic sequence?

Accepted Answer

a + (n - 1)d

Answer

a * r^(n - 1)

Answer

a + r^n

Answer

a * r + n

Question 20

Given the arithmetic sequence 2, 5, 8, 11, ..., what is the common difference?

Accepted Answer

3

Answer

2

Answer

4

Answer

5

Question 21

Which formula calculates the sum of the first n terms of an arithmetic sequence?

Accepted Answer

S = n(a + l)/2

Answer

S = n(a - l)/2

Answer

S = a + nl

Answer

S = (a + l) / n

Question 22

Which formula represents the nth term of a geometric sequence?

Accepted Answer

a * r^(n - 1)

Answer

a + (n - 1)b

Answer

a + r^n

Answer

a * r + n

Question 23

Which of the following statements defines the divergence test?

Accepted Answer

If the sequence is not bounded, then it diverges.

Answer

If the limit of the sequence is zero, then the sequence converges to zero.

Answer

If the limit of the sequence exists, then the sequence converges.

Answer

If the sequence is bounded, then it converges.

Question 24

What is the formula to calculate the nth term of an arithmetic sequence?

Accepted Answer

an = a1 + (n-1)d

Answer

an = a1 - (n-1)d

Answer

an = a1 * (n-1)d

Answer

an = a1 / (n-1)d

Question 25

Does the series 1 + 1/2 + 1/4 + 1/8 + ... converge or diverge?

Accepted Answer

Converges

Answer

Diverges

Answer

Cannot be determined

Question 26

Calculate the sum of the first 10 terms of the series: 1/2 + 1/4 + 1/8 + 1/16 + ....

Accepted Answer

1

Answer

1/2

Answer

1/4

Answer

1/8

Question 27

Which of the following is an example of a telescopic series?

Accepted Answer

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

Answer

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

Answer

1 + 2 + 4 + 8 + ...

Answer

1 - 2 + 4 - 8 + ...

Question 28

Which test is appropriate for determining if an alternating series is convergent?

Accepted Answer

Alternating series test

Answer

Ratio test

Answer

Integral test

Answer

Comparison test

Question 29

Which property characterizes Cauchy sequences?

Accepted Answer

They are convergent.

Answer

They are bounded.

Answer

They are monotonic.

Answer

They are unique.

Question 30

Find the nth term of the sequence: 3, 7, 11, 15, ...

Accepted Answer

2n + 1

Answer

n + 4

Answer

3n - 2

Answer

4n - 1

Question 31

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

Accepted Answer

Divergent

Answer

Convergent

Answer

Cannot be determined

Question 32

State and explain the necessary condition for the convergence of an infinite series. Provide an example to illustrate your explanation.

Accepted Answer

The nth term approaches zero.

Answer

The nth term is positive for all n.

Answer

The nth term is bounded.

Answer

The nth term is alternating in sign.

Question 33

Find the limit of the sequence a_n = (n^2 + 1)/(n^2 - 1) as n approaches infinity. Explain your method and provide a clear justification for your answer.

Accepted Answer

1

Answer

0

Answer

2

Answer

Infinity

Question 34

Which of the following sequences is a geometric sequence?

Accepted Answer

2, 4, 8, 16, 32

Answer

1, 3, 5, 7, 9

Answer

2, 3, 5, 8, 13

Answer

1, 2, 4, 8, 16

Question 35

What is the sum of the first 10 terms of the arithmetic sequence 5, 8, 11, 14, ...?

Accepted Answer

175

Answer

120

Answer

260

Answer

240

Question 36

Which of the following infinite series converges?

Accepted Answer

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

Answer

1 + 2 + 4 + 8 + ...

Answer

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

Answer

1 + 1/3 + 1/5 + 1/7 + ...

Question 37

What is the limit of the sequence a_n = (n^2 - 1) / (n^2 + 1) as n approaches infinity?

Accepted Answer

1

Answer

-1

Answer

0

Answer

Does not exist

Question 38

Which of the following series is a telescoping series?

Accepted Answer

1/(n+1) - 1/n

Answer

n^2 + n + 1

Answer

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

Answer

1/n + 1/n^2

Question 39

What is the common ratio of the geometric sequence 2, 6, 18, 54, ...?

Accepted Answer

3

Answer

2

Answer

6

Answer

4

Question 40

Which of the following geometric series converges?

Accepted Answer

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

Answer

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

Answer

1 + 2 + 4 + 8 + ...

Answer

1 - 2 + 4 - 8 + ...

Question 41

What is the nth term of the arithmetic sequence 3, 7, 11, 15, ...?

Accepted Answer

4n - 1

Answer

3n + 1

Answer

3n - 1

Answer

4n + 1

Question 42

Which of the following sequences is bounded?

Accepted Answer

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

Answer

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

Answer

1, 2, 3, 4, ...

Answer

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

Question 43

What is the sum of the first n terms of the geometric series 2 + 4 + 8 + 16 + ...?

Accepted Answer

2(2^n - 1)

Answer

2^n - 1

Answer

n(2^n + 1) / (2 + 1)

Answer

n(2^n - 1) / (2 - 1)

Question 44

Which of the following is a geometric sequence?

Accepted Answer

2, 4, 8, 16, ...

Answer

2, 3, 5, 7, ...

Answer

1, 4, 9, 16, ...

Answer

1, 2, 3, 4, ...

Question 45

What is the limit of the sequence 1/2, 1/4, 1/8, 1/16, ...?

Accepted Answer

0

Answer

Infinity

Answer

1/2

Answer

1

Question 46

Which of the following series converges?

Accepted Answer

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

Answer

1 - 2 + 4 - 8 + ...

Answer

1 + 2 + 4 + 8 + ...

Answer

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

Question 47

What is the sum of the first 10 terms of the arithmetic series 2, 5, 8, ...?

Accepted Answer

340

Answer

345

Answer

355

Answer

350

Question 48

Which of the following is a telescoping series?

Accepted Answer

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

Answer

1 + 2 + 4 + 8 + ...

Answer

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

Answer

1 - 1/2 + 1/4 - 1/8 + 1/16 - ...

Question 49

What is the general term of the geometric sequence 2, 6, 18, ...?

Accepted Answer

2 * 3^n

Answer

3 * 2^n

Answer

2 * 2^n

Answer

3 * 3^n

Question 50

Which of the following is a convergent alternating series?

Accepted Answer

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

Answer

1 + 2 + 4 + 8 + ...

Answer

1 - 1/2 + 1/4 - 1/8 + 1/16 - ...

Answer

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

Question 51

What is the sum of the first 5 terms of the arithmetic series 3, 7, 11, ...?

Accepted Answer

80

Answer

95

Answer

90

Answer

85

Question 52

Which of the following is a geometric series?

Accepted Answer

1 + 2 + 4 + 8 + ...

Answer

1, 4, 9, 16, ...

Answer

2, 3, 5, 7, ...

Answer

1, 2, 3, 4, ...

Question 53

What is the limit of the sequence 1, 1/2, 1/4, 1/8, ...?

Accepted Answer

0

Answer

Infinity

Answer

1/2

Answer

1