Series with Arbitrary Signs Practice Quiz (12 questions)

Question 1

1. For the sequence $u_n=\frac{(-1)^n(n+2)}{n+1}$ , which sequence gives the magnitudes in its alternating-series representation?

a_n=(-1)^n

a_n=\frac{n+1}{n+2}

a_n=\frac{(-1)^n(n+2)}{n+1}

a_n=\frac{n+2}{n+1}

Explanation

The positive magnitude sequence is uniquely determined by $a_n=|u_n|$ , which gives $a_n=\frac{n+2}{n+1}$ . Keeping the sign, taking the reciprocal, or using only the sign factor does not give the absolute values of the terms.

Answer

$$a_n=\frac{n+2}{n+1}$$

Question 2

2. A real sequence has terms $u_0=5$ , $u_1=-3$ , and $u_2=2$ . What sign must $u_3$ have if the sequence is alternating?

Either sign

Zero

Negative

Positive

Explanation

Consecutive terms of an alternating sequence have opposite signs, so the term after the negative $u_2$ must be positive. Repeating the preceding sign or comparing only terms of the same parity misses the sign change at each step.

Answer

The series converges.

Answer

$$-0.04\leq R_n<0$$

Answer

$$\alpha>0$$

Answer

The term magnitudes do not tend to zero

Answer

When $$\sum |u_n|$$ converges

Answer

The series $$\sum u_n$$ converges and its sum has absolute value at most $$5$$

Answer

Conditionally convergent

Answer

The series is conditionally convergent

Answer

The series diverges by the term test

Answer

Conditionally convergent

Question 3

3. A series has terms $u_n=(-1)^n a_n$ , where $a_n\geq 0$ decreases for all sufficiently large $n$ and tends to zero. What conclusion does the Alternating Series Test give?

The test gives no conclusion unless the magnitudes decrease from the first term.

The series converges.

The series converges absolutely.

The series diverges because the magnitudes decrease.

Explanation

The Alternating Series Test requires nonnegative magnitudes that are eventually decreasing and tend to zero; these conditions guarantee convergence. Eventual decrease is sufficient, and the test establishes convergence rather than absolute convergence.

Question 4

4. For a convergent alternating series, suppose the first omitted term after a partial sum is $u_{n+1}=-0.04$ . Which statement describes the remainder $R_n$ ?

|R_n|=|u_n|

R_n=-0.04

0<R_n\leq 0.04

-0.04\leq R_n<0

Explanation

The remainder has the same sign as the first omitted term, and its magnitude is at most that term's magnitude; hence $-0.04\leq R_n<0$ . A positive remainder reverses the sign, while equality with the omitted term is not guaranteed.

Question 5

5. For which values of $\alpha$ does the alternating series $\sum_{n\geq1}\frac{(-1)^n}{n^\alpha}$ converge?

\alpha>0

\alpha>1

\alpha\geq0

\alpha\leq0

Explanation

The alternating Riemann series converges exactly when $\alpha>0$ . The threshold $\alpha>1$ belongs to the standard Riemann series, while $\alpha\leq0$ fails the term test and $\alpha=0$ is not included.

Question 6

6. An alternating Riemann series has $\alpha=-0.4$ . What explains its divergence?

The term magnitudes do not tend to zero

The exponent is below the standard threshold

The alternating signs prevent cancellation

The term magnitudes decrease to zero

Explanation

For $\alpha\leq0$ , the term magnitude is $n^{-\alpha}\geq1$ , so the terms do not tend to zero and the series diverges by the term test. Alternation does not override this necessary condition; the standard Riemann threshold concerns a different series.

Question 7

7. When is a numerical series $\sum u_n$ called absolutely convergent?

When every

u_n

is nonnegative

When

\sum |u_n|

converges

When

\sum u_n

converges

When the partial sums of

\sum |u_n|

are bounded

Explanation

Absolute convergence is defined by convergence of the series of absolute values, $\sum |u_n|$ . Convergence of $\sum u_n$ alone does not give this definition, and nonnegative terms or bounded partial sums do not state the required criterion.

Question 8

8. A series has $\sum_{n=0}^{\infty}|u_n|=5$ . Which conclusion is guaranteed?

The series

\sum u_n

converges and its sum has absolute value at least

5

The series

\sum u_n

converges and its sum has absolute value exactly

5

The series

\sum u_n

diverges and its sum has absolute value at most

5

The series

\sum u_n

converges and its sum has absolute value at most

5

Explanation

Question 9

9. A series $\sum u_n$ converges, while the series of absolute values $\sum |u_n|$ diverges. How is $\sum u_n$ classified?

Absolutely convergent

Inconclusive by the term test

Divergent

Conditionally convergent

Explanation

Conditional convergence means the original series converges but its series of absolute values diverges. Absolute convergence requires the latter series to converge, while divergence of absolute values alone does not imply that the original series diverges.

Question 10

10. For $\sum_{n\geq1} \frac{(-1)^n}{n}$ , the Alternating Series Test establishes convergence, while the corresponding series of absolute values is the harmonic series. What follows?

The series is absolutely convergent

The term test is inconclusive about whether it converges

The series is conditionally convergent

The series is divergent

Explanation

The alternating harmonic series converges by the Alternating Series Test, but its absolute-value series $\sum_{n\geq1}1/n$ diverges, so it is conditionally convergent. Absolute convergence requires convergence of that harmonic series; its divergence does not establish divergence of the original series.

Question 11

11. A student is analyzing a series with arbitrary signs. Its terms do not tend to zero. What should the student conclude in the decision strategy?

The series requires an asymptotic expansion before any conclusion

The series diverges by the term test

The series converges absolutely if its signs alternate

The series converges if its absolute-value series diverges

Explanation

The decision strategy begins by checking whether $u_n\to0$ ; if not, the series diverges by the term test. Testing absolute convergence, applying the Alternating Series Test, or using an asymptotic expansion comes later in the procedure.

Question 12

12. For $u_n=(-1)^n(\sqrt{n+1}-\sqrt n)$ , rationalization shows that $|u_n|\sim 1/(2\sqrt n)$ . Since the magnitudes decrease to zero, how should the series be classified?

Inconclusive because the magnitudes decrease

Conditionally convergent

Absolutely convergent

Divergent by the term test

Explanation

The magnitudes decrease to zero, so the alternating series converges by the Alternating Series Test. Their asymptotic behavior is comparable to $1/(2\sqrt n)$ , whose series diverges, so absolute convergence fails and the series is conditional.

Series with Arbitrary Signs Practice Quiz (12 questions)

All questions and answers

1. For the sequence $u_n=\frac{(-1)^n(n+2)}{n+1}$ , which sequence gives the magnitudes in its alternating-series representation?

2. A real sequence has terms $u_0=5$ , $u_1=-3$ , and $u_2=2$ . What sign must $u_3$ have if the sequence is alternating?

3. A series has terms $u_n=(-1)^n a_n$ , where $a_n\geq 0$ decreases for all sufficiently large $n$ and tends to zero. What conclusion does the Alternating Series Test give?

4. For a convergent alternating series, suppose the first omitted term after a partial sum is $u_{n+1}=-0.04$ . Which statement describes the remainder $R_n$ ?

5. For which values of $\alpha$ does the alternating series $\sum_{n\geq1}\frac{(-1)^n}{n^\alpha}$ converge?

6. An alternating Riemann series has $\alpha=-0.4$ . What explains its divergence?

7. When is a numerical series $\sum u_n$ called absolutely convergent?

8. A series has $\sum_{n=0}^{\infty}|u_n|=5$ . Which conclusion is guaranteed?

9. A series $\sum u_n$ converges, while the series of absolute values $\sum |u_n|$ diverges. How is $\sum u_n$ classified?

10. For $\sum_{n\geq1} \frac{(-1)^n}{n}$ , the Alternating Series Test establishes convergence, while the corresponding series of absolute values is the harmonic series. What follows?

11. A student is analyzing a series with arbitrary signs. Its terms do not tend to zero. What should the student conclude in the decision strategy?

12. For $u_n=(-1)^n(\sqrt{n+1}-\sqrt n)$ , rationalization shows that $|u_n|\sim 1/(2\sqrt n)$ . Since the magnitudes decrease to zero, how should the series be classified?

Study with flashcards

Read the study guide

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Make your own practice quizzes

Series with Arbitrary Signs Practice Quiz (12 questions)

All questions and answers

1. For the sequence un=(−1)n(n+2)n+1u_n=\frac{(-1)^n(n+2)}{n+1}un​=n+1(−1)n(n+2)​, which sequence gives the magnitudes in its alternating-series representation?

2. A real sequence has terms u0=5u_0=5u0​=5, u1=−3u_1=-3u1​=−3, and u2=2u_2=2u2​=2. What sign must u3u_3u3​ have if the sequence is alternating?

3. A series has terms un=(−1)nanu_n=(-1)^n a_nun​=(−1)nan​, where an≥0a_n\geq 0an​≥0 decreases for all sufficiently large nnn and tends to zero. What conclusion does the Alternating Series Test give?

4. For a convergent alternating series, suppose the first omitted term after a partial sum is un+1=−0.04u_{n+1}=-0.04un+1​=−0.04. Which statement describes the remainder RnR_nRn​?

5. For which values of α\alphaα does the alternating series ∑n≥1(−1)nnα\sum_{n\geq1}\frac{(-1)^n}{n^\alpha}∑n≥1​nα(−1)n​ converge?

6. An alternating Riemann series has α=−0.4\alpha=-0.4α=−0.4. What explains its divergence?

7. When is a numerical series ∑un\sum u_n∑un​ called absolutely convergent?

8. A series has ∑n=0∞∣un∣=5\sum_{n=0}^{\infty}|u_n|=5∑n=0∞​∣un​∣=5. Which conclusion is guaranteed?

9. A series ∑un\sum u_n∑un​ converges, while the series of absolute values ∑∣un∣\sum |u_n|∑∣un​∣ diverges. How is ∑un\sum u_n∑un​ classified?

10. For ∑n≥1(−1)nn\sum_{n\geq1} \frac{(-1)^n}{n}∑n≥1​n(−1)n​, the Alternating Series Test establishes convergence, while the corresponding series of absolute values is the harmonic series. What follows?

11. A student is analyzing a series with arbitrary signs. Its terms do not tend to zero. What should the student conclude in the decision strategy?

12. For un=(−1)n(n+1−n)u_n=(-1)^n(\sqrt{n+1}-\sqrt n)un​=(−1)n(n+1​−n​), rationalization shows that ∣un∣∼1/(2n)|u_n|\sim 1/(2\sqrt n)∣un​∣∼1/(2n​). Since the magnitudes decrease to zero, how should the series be classified?

Study with flashcards

Read the study guide

Related classes

Oomycete Biology and Life Cycles

Periodic Table and Periodicity

Aldehydes Ketones and Carboxylic Acids

Physics Foundations and Motion

Variables and Hypotheses

Weighted Averages of Points

Make your own practice quizzes

1. For the sequence $u_n=\frac{(-1)^n(n+2)}{n+1}$ , which sequence gives the magnitudes in its alternating-series representation?

2. A real sequence has terms $u_0=5$ , $u_1=-3$ , and $u_2=2$ . What sign must $u_3$ have if the sequence is alternating?

3. A series has terms $u_n=(-1)^n a_n$ , where $a_n\geq 0$ decreases for all sufficiently large $n$ and tends to zero. What conclusion does the Alternating Series Test give?

4. For a convergent alternating series, suppose the first omitted term after a partial sum is $u_{n+1}=-0.04$ . Which statement describes the remainder $R_n$ ?

5. For which values of $\alpha$ does the alternating series $\sum_{n\geq1}\frac{(-1)^n}{n^\alpha}$ converge?

6. An alternating Riemann series has $\alpha=-0.4$ . What explains its divergence?

7. When is a numerical series $\sum u_n$ called absolutely convergent?

8. A series has $\sum_{n=0}^{\infty}|u_n|=5$ . Which conclusion is guaranteed?

9. A series $\sum u_n$ converges, while the series of absolute values $\sum |u_n|$ diverges. How is $\sum u_n$ classified?

10. For $\sum_{n\geq1} \frac{(-1)^n}{n}$ , the Alternating Series Test establishes convergence, while the corresponding series of absolute values is the harmonic series. What follows?

12. For $u_n=(-1)^n(\sqrt{n+1}-\sqrt n)$ , rationalization shows that $|u_n|\sim 1/(2\sqrt n)$ . Since the magnitudes decrease to zero, how should the series be classified?