Master Infinite Sums: Conquer Series in Calculus II

Question 1

Which convergence test could NOT be used for the series ∑(n=1 to ∞) n / (n^2 + 1)?

Accepted Answer

Alternating Series Test

Answer

Integral Test

Answer

Ratio Test

Answer

Comparison Test

Question 2

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

Accepted Answer

Converges to 0

Answer

Converges to 1

Answer

Diverges to infinity

Answer

Diverges to negative infinity

Question 3

Determine the Maclaurin series expansion for sin(x).

Accepted Answer

∑(n=0 to ∞) (-1)^n * x^(2n + 1) / (2n + 1)!

Answer

∑(n=0 to ∞) x^(2n + 1) / (2n + 1)!

Answer

∑(n=1 to ∞) (-1)^n * x^n / n!

Answer

∑(n=1 to ∞) x^n / n!

Question 4

Apply the appropriate convergence test to determine the convergence or divergence of the series ∑(n=2 to ∞) (ln(n)) / n.

Accepted Answer

Limit Comparison Test

Answer

Integral Test

Answer

Ratio Test

Answer

Comparison Test

Question 5

On which mathematical criterion are all convergence tests based?

Accepted Answer

Cauchy criterion for convergence of a sequence

Answer

Limit comparison test

Answer

Weierstrass M-test

Question 6

Which condition is necessary for a series to converge?

Accepted Answer

The terms of the series approach zero as n approaches infinity

Answer

The series must have a positive sum

Answer

The terms of the series alternate in sign

Answer

The series must have a finite number of terms

Question 7

Which convergence test is appropriate for the series 1 + 1/2 + 1/4 + 1/8 + ...?

Accepted Answer

Ratio Test

Answer

Integral Test

Answer

Alternating Series Test

Answer

Comparison Test

Question 8

Determine whether the series converges or diverges:

$$sum_{n=1}^{infty} rac{1}{n^2}$$

Accepted Answer

Converges

Answer

Diverges

Question 9

Which of the following is NOT a type of convergence test for series?

Accepted Answer

Multiplication Test

Answer

Comparison Test

Answer

Integral Test

Question 10

Determine whether the alternating series:

$$sum_{n=1}^{infty} (-1)^n sqrt{rac{1}{n}}$$

converges absolutely, conditionally, or diverges.

Accepted Answer

Converges conditionally

Answer

Converges absolutely

Answer

Diverges

Question 11

Find the Maclaurin series for f(x) = sin(x).

Accepted Answer

$$-sum_{n=0}^{infty} rac{(-1)^n}{2n+1}x^{2n+1}$$

Answer

$$sum_{n=0}^{infty} rac{x^n}{n!}$$

Answer

$$-sum_{n=0}^{infty} rac{(-1)^n}{2n}x^{2n}$$

Question 12

Which of the following functions is NOT represented by a power series?

Accepted Answer

$x^2 + 1$

Answer

$e^x$

Answer

sin(x)

Question 13

Determine the convergence of the series:

$$sum_{n=1}^{infty} rac{ln(n)}{n}$$

Accepted Answer

Diverges

Answer

Converges

Question 14

Find the sum of the series:

$$sum_{n=1}^{infty} rac{1}{2^n + 3^n}$$

Accepted Answer

1/5

Answer

1/4

Answer

1/3

Question 15

Which of the following is the correct definition of a series?

Accepted Answer

An infinite sum of numbers

Answer

A finite sum of numbers

Answer

A polynomial divided by a power of x

Answer

A sequence of numbers that converges to a limit

Question 16

Evaluate the effectiveness of the unit's assessment in measuring students' understanding of series concepts.

Accepted Answer

The assessment thoroughly evaluates students' grasp of series concepts and aligns with learning objectives.

Answer

The assessment overemphasizes memorization and neglects higher-order thinking skills.

Answer

The assessment is excessively challenging and hinders students from fairly demonstrating their understanding.

Answer

The assessment is too narrow and fails to cover the range of series concepts taught in the unit.

Question 17

Distinguish between conditional convergence, absolute convergence, and divergence in the context of infinite series.

Accepted Answer

Conditional convergence: Σa_n converges, but Σ|a_n| diverges.

Accepted Answer

Absolute convergence: Both Σa_n and Σ|a_n| converge.

Accepted Answer

Divergence: Both Σa_n and Σ|a_n| diverge.

Answer

Conditional convergence: Only Σ|a_n| diverges.

Question 18

Determine the convergent series from the given options:

Accepted Answer

∑ (1/n^2)

Answer

∑ n

Answer

∑ (n^2)

Answer

∑ (1/n)

Question 19

Which test is specifically designed to determine the convergence of an alternating series?

Accepted Answer

The Alternating Series Test

Answer

The Comparison Test

Answer

The Ratio Test

Answer

The Integral Test

Question 20

Identify the geometric series from the provided options:

Accepted Answer

∑ (0.5)^n

Answer

∑ (n^3)

Answer

∑ (1/n!)

Answer

∑ (n+1)

Question 21

Provide the correct formula for the nth term of a power series:

Accepted Answer

a_n = f^(n)(a) / n!

Answer

a_n = f^(n)(0) / n

Answer

a_n = f'(a)^n / n

Answer

a_n = f(a)^n / n!

Question 22

Identify the telescoping series from the given options:

Accepted Answer

∑ (1/(n+1) - 1/n)

Answer

∑ (1/n^2)

Answer

∑ n

Answer

∑ (n^3)

Question 23

Which test is specifically designed for determining the convergence or divergence of a p-series?

Accepted Answer

The p-Series Test

Answer

The Ratio Test

Answer

The Integral Test

Question 24

Which of the following is a power series representation of the function f(x) = e^x?

Accepted Answer

∑(x^n/n!) for all x

Answer

∑(x^n/n) for all x

Answer

∑(1/n^x) for all x

Answer

∑(e^n/n!) for all x

Question 25

Find the limit of the sequence {a_n} where a_n = ∑(1/k^2) for k = 1 to n.

Accepted Answer

π^2/6

Answer

1

Answer

∞

Answer

0

Question 26

Consider the series ∑(n=1 to ∞) (-1)^n / sqrt(n). Which statement about this series is true?

Accepted Answer

It is an alternating series that converges.

Answer

It is an alternating series that diverges.

Answer

It is a geometric series that converges.

Answer

It is a geometric series that diverges.

Question 27

Which of the following series is a geometric series?

Accepted Answer

∑(n=0 to ∞) (1/2)^n

Answer

∑(n=1 to ∞) (-1)^n

Answer

∑(n=1 to ∞) n^2

Answer

∑(n=1 to ∞) 1 / n

Question 28

Determine the convergence of the series ∑ (n=1 to ∞) (1 / n^2)

Accepted Answer

Converges

Answer

Converges conditionally

Answer

Diverges

Answer

Cannot be determined

Question 29

Use the Ratio Test to determine the convergence or divergence of the series ∑ (n=1 to ∞) ((2^n + 5) / (3^n - 1))

Accepted Answer

Converges

Answer

Diverges

Answer

Cannot be determined

Question 30

Find the Maclaurin series for the function f(x) = ln(1 + x)

Accepted Answer

∑ (n=1 to ∞) ((-1)^(n+1) * x^n) / n

Answer

∑ (n=1 to ∞) (x^n / n)

Answer

∑ (n=1 to ∞) ((-1)^n * x^n) / n

Answer

∑ (n=1 to ∞) ((x^n) / (n+1))

Question 31

Which of the following is a telescoping series?

Accepted Answer

∑ (n=1 to ∞) (1 / (n+1) - 1 / n)

Answer

∑ (n=1 to ∞) (1 / n)

Answer

∑ (n=1 to ∞) (1 / n^2)

Answer

∑ (n=1 to ∞) (n / (n+1))

Question 32

Use the Limit Comparison Test to determine the convergence or divergence of the series: 1 + 1/(e^2) + 1/(e^4) + 1/(e^6) + ...

Accepted Answer

Converges

Answer

Diverges

Answer

Cannot be determined

Question 33

Use the Integral Test to determine whether the series 1/2 + 1/4 + 1/8 + 1/16 + ... converges or diverges.

Accepted Answer

Converges

Answer

Cannot be determined

Answer

Converges conditionally

Answer

Diverges

Question 34

Use the Comparison Test to determine the convergence or divergence of the series: 1 + 1/sqrt(2) + 1/(sqrt(2))^2 + 1/(sqrt(2))^3 + ...

Accepted Answer

Converges

Answer

Converges conditionally

Answer

Diverges

Answer

Cannot be determined

Question 35

Determine whether the series ∑(n=1 to ∞) (n / (n + 1)) converges or diverges. Apply the limit comparison test with ∑(n=1 to ∞) 1/n

Accepted Answer

Converges

Answer

Diverges

Answer

Converges conditionally

Answer

Diverges conditionally

Question 36

Which of the following series is a geometric series?

Accepted Answer

∑(n=1 to ∞) (2 / 3)^n

Answer

∑(n=1 to ∞) n^2

Answer

∑(n=1 to ∞) n

Answer

∑(n=1 to ∞) 1 / n

Question 37

Find the sum of the first 5 terms of the series ∑(n=1 to ∞) n / (2^n).

Accepted Answer

31 / 16

Answer

2

Answer

15 / 16

Answer

1 / 2

Question 38

Which of the following series is represented by the power series ∑(n=0 to ∞) (x^n / n!)?

Accepted Answer

e^x

Answer

cos(x)

Answer

sin(x)

Answer

ln(x)

Question 39

Determine whether the series ∑(n=2 to ∞) (1 / ln(n)^2) converges or diverges. Apply the integral test for convergence.

Accepted Answer

Converges

Answer

Diverges

Answer

Diverges conditionally

Answer

Converges conditionally

Question 40

Find the sum of the series ∑(n=1 to ∞) (1 / (n(n + 1)))

Accepted Answer

1

Answer

0

Answer

2

Answer

∞

Question 41

Which of the following series converges?

Accepted Answer

Σ (n=1 to ∞) 1/(n^2)

Answer

Σ (n=1 to ∞) n

Answer

Σ (n=1 to ∞) 1/n

Answer

Σ (n=1 to ∞) (-1)^n

Question 42

Using the Ratio Test, determine if the series Σ (n=1 to ∞) ((-1)^n * n^2) / (2n)^n converges.

Accepted Answer

Converges

Answer

None of the above

Answer

Diverges

Answer

Cannot be determined

Question 43

Identify the geometric series from the following options.

Accepted Answer

Σ (n=1 to ∞) 2^n

Answer

Σ (n=1 to ∞) 1/n^2

Answer

Σ (n=1 to ∞) (-1)^n / (2n+1)

Answer

Σ (n=1 to ∞) n / (n+1)

Question 44

Which of the following series represents a power series?

Accepted Answer

Σ (n=0 to ∞) (-1)^n * x^n / n!

Answer

Σ (n=1 to ∞) n

Answer

Σ (n=1 to ∞) 1/n^2

Answer

Σ (n=1 to ∞) (-1)^n

Question 45

The interval of convergence for the power series Σ (n=1 to ∞) (x-2)^n / n is (-∞, ∞).

Accepted Answer

False

Answer

Cannot be determined

Answer

True

Answer

None of the above

Question 46

Which of the following is a necessary condition for a power series to converge?

Accepted Answer

|x| < R

Answer

x > R

Answer

x < R

Answer

|x| > R

Question 47

Determine the radius of convergence for the power series Σ (n=1 to ∞) (x+1)^n / n.

Accepted Answer

1

Answer

0

Answer

-1

Answer

∞

Question 48

What is the nature of the derivative of a power series?

Accepted Answer

Term-by-term differentiable

Answer

Cannot be differentiated

Answer

Not term-by-term differentiable

Answer

Only differentiable at certain points

Question 49

What is the sum of the infinite geometric series: $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...$ ?

Accepted Answer

2

Answer

1

Answer

4

Answer

The series diverges.

Question 50

Which test can be used to determine the convergence of the series $\sum_{n=2}^\infty \frac{1}{n \ln(n)}$?

Accepted Answer

Integral Test

Answer

Comparison Test

Answer

Ratio Test

Question 51

The series $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$ is:

Accepted Answer

Conditionally Convergent

Answer

Divergent

Answer

Absolutely Convergent

Answer

Harmonic

Question 52

What is the radius of convergence for the power series $\sum_{n=0}^\infty \frac{x^n}{n!}$?

Accepted Answer

Infinity

Answer

0

Answer

-1

Answer

1

Question 53

The Taylor series expansion of $e^x$ centered at x=0 is:

Accepted Answer

$1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$

Answer

$1 + \frac{x^2}{2!} + \frac{x^4}{4!} + ...$

Answer

$1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + ...$

Answer

$x + \frac{x^2}{2} + \frac{x^3}{3} + ...$

Question 54

If a series is convergent, then:

Accepted Answer

The limit of the nth term as n approaches infinity must be 0.

Answer

The terms of the series are always positive.

Answer

The series is also absolutely convergent.

Question 55

Which of the following is a valid representation of the function $f(x) = \frac{1}{1-x}$ as a power series centered at x=0?

Accepted Answer

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

Answer

$1 + \frac{x}{2} + \frac{x^2}{3} + \frac{x^3}{4} + ...$

Answer

$1 - x + x^2 - x^3 + ...$

Answer

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

Question 56

A series of the form $\sum_{n=1}^\infty \frac{1}{n^p}$ is convergent if:

Accepted Answer

p > 1

Answer

p < 1

Answer

p = 1