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Free Practice Quiz Question List

Sequences and Infinite Series Online Quiz Questions

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

20 questions
01
Choose one
1 point

Which statement correctly defines convergence of an infinite series?

  1. A

    A series converges when its terms eventually become exactly zero.

  2. B

    A series converges when its sequence of partial sums approaches a finite limit.

  3. C

    A series converges whenever its terms alternate in sign.

  4. D

    A series converges whenever its terms are bounded.

02
Fill in the blank
1 point

A formula that gives a sequence term directly from its index is an definition, while a definition that supplies initial terms and a rule for obtaining later terms is a definition.

03
True or false
1 point

True or false: Adding or removing finitely many terms from a series can change its convergence classification.

  1. A

    True

  2. B

    False

04
Choose all
1 point

For the Ratio Test, let L=lim⁡n→∞∣an+1an∣L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|. Which conclusions are valid? Select all correct choices.

  1. A

    L<1L<1 implies absolute convergence.

  2. B

    L>1L>1 implies divergence.

  3. C

    L=1L=1 implies convergence.

  4. D

    L=∞L=\infty implies divergence.

05
Choose one
1 point

After the Ratio Test identifies the radius of convergence of a power series, what must be done at the endpoints?

  1. A

    The Ratio Test always proves convergence at both endpoints.

  2. B

    The endpoints can be classified from the radius alone without further work.

  3. C

    Each endpoint must be tested separately.

  4. D

    Power series have no endpoint cases.

06
Choose one
1 point

Which set of conditions allows the Integral Test to relate a positive-term series ∑n=N∞an\sum_{n=N}^{\infty}a_n to an improper integral?

  1. A

    The function must be negative, continuous, and increasing, with f(n)=anf(n)=a_n.

  2. B

    The function must be positive, continuous, and eventually decreasing, with f(n)=anf(n)=a_n.

  3. C

    The function must be periodic and bounded, with f(n)=anf(n)=a_n.

  4. D

    The function must be a polynomial, with f(n)=anf(n)=a_n.

07
Choose one
1 point

What is the sum of the geometric series ∑n=0∞6(12)n\sum_{n=0}^{\infty}6\left(\frac{1}{2}\right)^n?

  1. A

    6

  2. B

    12

  3. C

    3

  4. D

    Does not converge

08
True or false
1 point

True or false: The series ∑n=1∞nn+1\sum_{n=1}^{\infty}\frac{n}{n+1} diverges because its terms do not approach zero.

  1. A

    True

  2. B

    False

09
Choose one
1 point

What is the sum of the telescoping series ∑n=1∞1n(n+1)\sum_{n=1}^{\infty}\frac{1}{n(n+1)}?

  1. A

    0

  2. B

    12\frac{1}{2}

  3. C

    1

  4. D

    The series diverges

10
Choose one
1 point

For the alternating harmonic series ∑n=1∞(−1)n−11n\sum_{n=1}^{\infty}(-1)^{n-1}\frac{1}{n}, what is the greatest error bound guaranteed after using the first 8 terms?

  1. A

    19\frac{1}{9}

  2. B

    18\frac{1}{8}

  3. C

    110\frac{1}{10}

  4. D

    1

11
Choose one
1 point

Classify the series ∑n=1∞2n2+1n3+4\sum_{n=1}^{\infty}\frac{2n^2+1}{n^3+4} using an appropriate comparison.

  1. A

    Converges absolutely

  2. B

    Converges conditionally

  3. C

    Converges by the Integral Test

  4. D

    Diverges

12
True or false
1 point

True or false: If a series converges absolutely, then it also converges ordinarily.

  1. A

    True

  2. B

    False

13
Written response
1 point

Enter the exact value of ∑n=0∞2(14)n\sum_{n=0}^{\infty}2\left(\frac{1}{4}\right)^n as a reduced fraction.

14
Choose one
1 point

Classify the positive-term series ∑n=1∞1n2+sin⁡2n\sum_{n=1}^{\infty}\frac{1}{n^2+\sin^2 n}.

  1. A

    Diverges by comparison with the harmonic series

  2. B

    Converges by Direct Comparison with ∑1/n2\sum 1/n^2

  3. C

    Converges conditionally by the Alternating Series Test

  4. D

    The nth-Term Test is inconclusive, so no test applies

15
Choose all
1 point

Which statements are required, or describe the required structure, for applying the Alternating Series Test to a series of the form ∑n=1∞(−1)n−1bn\sum_{n=1}^{\infty}(-1)^{n-1}b_n? Select all correct choices.

  1. A

    The magnitudes satisfy bn≥0b_n\ge 0.

  2. B

    The magnitudes are eventually decreasing.

  3. C

    The magnitudes approach zero: lim⁡n→∞bn=0\lim_{n\to\infty}b_n=0.

  4. D

    The signs of successive terms alternate.

  5. E

    The terms must have a constant positive ratio.

16
Written response
1 point

An increasing sequence is known to be bounded above. What theorem guarantees that the sequence converges?

17
Written response
1 point

Find the limit: lim⁡n→∞4n2−n+12n2+3\displaystyle \lim_{n\to\infty}\frac{4n^2-n+1}{2n^2+3}. Enter the limiting value as a number.

18
Written response
1 point

Find the limit: lim⁡n→∞n52n\displaystyle\lim_{n\to\infty}\frac{n^5}{2^n}. Enter the limiting value as a number.

19
Fill in the blank
1 point

To apply the Alternating Series Test to ∑n=1∞(−1)n−1bn\sum_{n=1}^{\infty}(-1)^{n-1}b_n, verify that bnb_n is and that lim⁡n→∞bn=\lim_{n\to\infty}b_n=.

20
Open ended
1 point

Determine whether ∑n=1∞(−1)n−11n\displaystyle\sum_{n=1}^{\infty}(-1)^{n-1}\frac{1}{n} converges. Use the Alternating Series Test and explain how each of its hypotheses applies.