Infinite Series: Convergence Tests and Properties - Interactive Learning Quizzes

Question 1

Consider the power series ∑ (x - 2)^n / n. What is the radius of convergence of this power series?

Accepted Answer

1

Answer

0

Answer

2

Answer

The series has no radius of convergence and diverges for all values of x.

Question 2

Consider the series ∑ (n^2 + 1) / (n^3 + 2n). Which of the following tests can be used to demonstrate that this series diverges?

Accepted Answer

Direct comparison test

Answer

Integral test

Answer

Ratio test

Answer

Alternating series test

Question 3

Which of the following series represents the Taylor series expansion of the function f(x) = e^x centered at x = 0?

Accepted Answer

∑ x^n / n!

Answer

∑ (-1)^n x^n / n!

Answer

∑ x^n / n

Answer

∑ (-1)^n x^n

Question 4

Which of the following is NOT a necessary condition for the Alternating Series Test?

Accepted Answer

The limit of the terms must be 0

Answer

The terms alternate in sign

Answer

The magnitude of the terms decreases as n increases

Question 5

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

Accepted Answer

∑((-1)^n * x^(2n+1) / (2n+1)!)

Answer

∑(x^n / n!)

Answer

∑(x^n / (2n+1)!)

Answer

∑((-1)^n * x^n / n!)

Question 6

Does the series ∑_(n=1)^∞ (1/n^3 + n/(n^2 + 1)) converge or diverge? Use the comparison test.

Accepted Answer

The series converges because 1/n^3 converges and n/(n^2 + 1) converges.

Answer

The series diverges because 1/n^3 diverges and n/(n^2 + 1) converges.

Answer

The series converges because the limit of the series as n approaches infinity is 0.

Question 7

Which of the following statements about alternating series is true?

Accepted Answer

If the terms of an alternating series decrease monotonically to zero, then the series is convergent.

Answer

Alternating series always converge.

Answer

Alternating series always diverge.

Question 8

Which of the following series is telescoping?

Accepted Answer

$\sum_{n=1}^{\infty}\frac{1}{n+1}-rac{1}{n+2}$

Answer

$\sum_{n=1}^{\infty}\frac{1}{n}$

Answer

$\sum_{n=1}^{\infty}\frac{n}{n+1}$

Answer

$\sum_{n=1}^{\infty}\frac{n^2}{n+1}$

Question 9

Determine whether the alternating series Σ (-1)^(n+1) / n^(1/2) is convergent or divergent.

Accepted Answer

Convergent

Answer

Divergent

Answer

None of the above

Answer

Conditional Convergent

Question 10

Identify the correct series representation for the function f(x) = e^x.

Accepted Answer

Σ x^n / n!

Answer

Σ (x-1)^n / n!

Answer

Σ cos(x)/n

Answer

Σ x^n / 2^n

Question 11

Determine whether the series ∑(n=1 to ∞) ((n^2 + 1) / (n^3 + 2n)) converges or diverges. Apply the ratio test.

Accepted Answer

Converges

Answer

Conditionally convergent

Answer

Absolutely convergent

Answer

Diverges

Question 12

Test the series ∑(n=1 to ∞) (-1)^n / n for absolute convergence using the alternating series test.

Accepted Answer

Absolutely convergent

Answer

Divergent

Answer

Convergent

Answer

Conditionally convergent

Question 13

Consider the series ∑(n=1 to ∞) [(−1)^n * (n^2 + 1)] / (n^3 + 2). Determine if it is conditionally convergent or absolutely convergent.

Accepted Answer

Conditionally convergent

Answer

Divergent

Answer

Convergent but not absolutely convergent

Answer

Absolutely convergent

Question 14

Determine the convergence or divergence of the series ∑(n=2 to ∞) (1 / (n * ln(n))) using an appropriate test.

Accepted Answer

Converges

Answer

Diverges

Answer

Absolutely convergent

Answer

Conditionally convergent

Question 15

Use the Ratio Test to determine whether the series $\sum_{n=1}^\infty \frac{(2n)!}{n^n}$ converges or diverges.

Accepted Answer

Diverges

Answer

Converges

Answer

Cannot be determined without using other tests

Answer

The series does not satisfy the conditions of the Ratio Test

Question 16

Determine if the series $\sum_{n=1}^\infty \left(\frac{1}{n} - \frac{1}{n+1}\right)$ converges or diverges.

Accepted Answer

Converges

Answer

Cannot be determined without using other tests

Answer

The test does not apply to series involving telescopic sums

Answer

Diverges

Question 17

Use the Alternating Series Test to determine whether the series $\sum_{n=1}^\infty (-1)^n \frac{n}{n^2 + 1}$ converges or diverges.

Accepted Answer

Converges

Answer

Cannot be determined using the Alternating Series Test

Answer

The test does not apply to series with positive terms

Answer

Diverges

Question 18

Consider the series $\sum_{n=1}^\infty \frac{(n+1)!}{n^n}$. Utilize the Root Test to assess its convergence or divergence.

Accepted Answer

Diverges

Answer

Converges

Answer

Cannot be determined without using other tests

Question 19

Utilize the Comparison Test to determine whether the series $\sum_{n=1}^\infty \frac{1}{n^{3/2}}$ converges or diverges.

Accepted Answer

Converges

Answer

The test does not apply to series involving fractional exponents

Answer

Cannot be determined without using other tests

Answer

Diverges

Question 20

Which of the following series is a geometric series?

Accepted Answer

∑(n=1 to ∞) (1/2)^n

Answer

∑(n=1 to ∞) (-1)^n

Answer

∑(n=1 to ∞) 1/n

Answer

∑(n=1 to ∞) n^2

Question 21

The series ∑(n=1 to ∞) (n + 1) / e^n is:

Accepted Answer

Absolutely convergent

Answer

Cannot be determined

Answer

Divergent

Answer

Conditionally convergent

Question 22

Which test can be used to determine that the series ∑(n=1 to ∞) (-1)^(n-1) / n is conditionally convergent?

Accepted Answer

Alternating series test

Answer

Ratio test

Answer

Integral test

Answer

Comparison test

Question 23

What is the limit of the sequence {a_n} if a_n = ∑(k=1 to n) (1 / k^2)?

Accepted Answer

π^2 / 6

Answer

1

Answer

π

Answer

0

Question 24

The series ∑(n=1 to ∞) sin(n) is:

Accepted Answer

Divergent

Answer

Convergent

Answer

Absolutely convergent

Answer

Conditionally convergent

Question 25

Which of the following series converges absolutely?

Accepted Answer

∑ (-1)^n / n^2

Answer

∑ (-1)^n / n

Answer

∑ (n+1)/n^2

Answer

∑ 1/n

Question 26

The series ∑ (-1)^n / (n+1) is:

Accepted Answer

Conditionally convergent

Answer

Divergent

Answer

Absolutely convergent

Answer

Convergent, but not absolutely convergent

Question 27

The limit of the sequence {a_n} = {(-1)^n / n} as n approaches infinity is:

Accepted Answer

0

Answer

1

Answer

Does not exist

Answer

-1

Question 28

If the limit as n approaches infinity of a_n in the series ∑ a_n is not 0, then the series:

Accepted Answer

Diverges

Answer

Converges absolutely

Answer

Converges conditionally

Answer

Converges

Question 29

Which of the following series can be tested for convergence using the root test?

Accepted Answer

∑ (n/2^n)

Answer

∑ 1/n

Answer

∑ (-1)^n / n

Answer

∑ (n+1)/n^2