Timedora
Free Practice Quiz Question List

6 Infinite Series and Convergence Tests Online Quiz Questions

Use this free practice quiz with 20 questions to review 6 Infinite Series and Convergence Tests, test your knowledge, and prepare for your next test or exam.

20 questions
01
Choose one
1 point

Which condition guarantees that the geometric series ∑n=0∞arn\sum_{n=0}^{\infty} ar^n converges?

  1. A

    A geometric series converges whenever r<1r<1.

  2. B

    A geometric series converges exactly when ∣r∣<1|r|<1.

  3. C

    A geometric series converges whenever r>0r>0.

  4. D

    A geometric series converges exactly when ∣r∣>1|r|>1.

02
Choose one
1 point

Evaluate the telescoping series ∑n=1∞1n(n+1)\sum_{n=1}^{\infty}\frac{1}{n(n+1)}.

  1. A

    1

  2. B

    0

  3. C

    12\frac{1}{2}

  4. D

    The series diverges

03
Choose one
1 point

What is the correct conclusion for ∑n=1∞1n3/2\sum_{n=1}^{\infty}\frac{1}{n^{3/2}}?

  1. A

    It diverges because p<2p<2.

  2. B

    It diverges because every pp-series diverges.

  3. C

    It converges because p>1p>1.

  4. D

    The pp-series test is inconclusive for this value.

04
Choose all
1 point

Which statements correctly describe comparison tests for series? Select all correct choices.

  1. A

    The direct comparison test is primarily used for nonnegative terms.

  2. B

    The limit comparison test applies when the limiting ratio is zero.

  3. C

    The limit comparison test applies when the limiting ratio lies strictly between 0 and infinity.

  4. D

    The direct comparison test requires the two series to have equal sums.

05
Choose all
1 point

Which conditions are required by the alternating series test? Select all correct choices.

  1. A

    The signs must eventually be positive.

  2. B

    The term magnitudes must eventually decrease.

  3. C

    The series of absolute values must converge.

  4. D

    The term magnitudes must approach zero.

06
True or false
1 point

True or false: If lim⁡n→∞an=0\lim_{n\to\infty}a_n=0, then the series ∑n=1∞an\sum_{n=1}^{\infty}a_n must converge.

  1. A

    True

  2. B

    False

07
True or false
1 point

True or false: If the ratio-test limit L=lim⁡n→∞∣an+1/an∣L=\lim_{n\to\infty}\left|a_{n+1}/a_n\right| is less than 1, then the series converges absolutely.

  1. A

    True

  2. B

    False

08
Written response
1 point

What is the sum of ∑n=1∞3(14)n−1\sum_{n=1}^{\infty}3\left(\frac{1}{4}\right)^{n-1}? Enter the numerical answer as a simplified decimal or fraction.

09
Written response
1 point

What is the convergence classification of ∑n=1∞(−1)n−1n\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n}? Enter one concise classification.

10
Fill in the blank
1 point

The harmonic series \sum_{n=}^{\infty}\frac{1}{n} has p=1p=1, so it .

11
Fill in the blank
1 point

Using the root test, ∑n=1∞(2n+15n)n\sum_{n=1}^{\infty}\left(\frac{2n+1}{5n}\right)^n because its root-test limit is 25\frac{2}{5}.

12
Open ended
1 point

Use the limit comparison test to determine whether ∑n=1∞3n2+1n3−2\sum_{n=1}^{\infty}\frac{3n^2+1}{n^3-2} converges or diverges. Show the comparison limit and justify your conclusion.

13
Choose one
1 point

For a series, the ratio-test limit is L=1L=1. What conclusion does the ratio test itself provide?

  1. A

    The series converges absolutely.

  2. B

    The ratio test is inconclusive.

  3. C

    The series diverges.

  4. D

    The series is conditionally convergent.

14
Choose one
1 point

If ∑n=1∞(bn−bn+1)\sum_{n=1}^{\infty}(b_n-b_{n+1}) has b1=5b_1=5 and lim⁡n→∞bn+1=2\lim_{n\to\infty}b_{n+1}=2, what is its sum?

  1. A

    b1+Lb_1+L

  2. B

    L−b1L-b_1

  3. C

    b1−Lb_1-L

  4. D

    The series always diverges

15
Choose one
1 point

Determine whether the geometric series ∑n=0∞4(−32)n\sum_{n=0}^{\infty}4\left(-\frac{3}{2}\right)^n converges or diverges.

  1. A

    The series converges because rr is positive.

  2. B

    The series diverges because ∣r∣>1|r|>1.

  3. C

    The series converges to −2-2.

  4. D

    The series diverges because its first term is negative.

16
Choose one
1 point

What conclusion follows for the series ∑n=1∞1n3/2\sum_{n=1}^{\infty}\frac{1}{n^{3/2}}?

  1. A

    It diverges because the exponent is less than 2.

  2. B

    It diverges because every reciprocal-power series diverges.

  3. C

    It converges because 32>1\frac{3}{2}>1.

  4. D

    It converges only if its first term is less than 1.

17
Choose one
1 point

Classify the series ∑n=1∞(−1)n−1n\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n}.

  1. A

    It converges by the alternating series test.

  2. B

    It diverges because the harmonic series is divergent.

  3. C

    It converges absolutely by the pp-series test.

  4. D

    It diverges because its terms alternate in sign.

18
True or false
1 point

True or false: If the ratio-test limit lim⁡n→∞∣an+1an∣\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| equals 11, the series must diverge.

  1. A

    True

  2. B

    False

19
Written response
1 point

Enter the sum of the geometric series ∑n=1∞3(14)n−1\sum_{n=1}^{\infty}3\left(\frac{1}{4}\right)^{n-1} as a simplified number.

20
Written response
1 point

For the series ∑n=1∞3n2+1n3−2\sum_{n=1}^{\infty}\frac{3n^2+1}{n^3-2}, enter whether the series converges or diverges.