Question 1

Which of the following statements best describes absolute convergence for a series $$\sum a_n$$?

Accepted Answer

The series $$\sum a_n$$ converges absolutely if the corresponding series of absolute values, $$\sum |a_n|$$, converges.

Question 2

According to Theorem 12 (The Absolute Convergence Test), what can be concluded if $$\sum_{n=1}^{\infty} |a_n|$$ converges?

Accepted Answer

The series $$\sum_{n=1}^{\infty} a_n$$ converges.

Question 3

Given the series $$\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$$, which of the following is true?

Accepted Answer

The series is conditionally convergent.

Question 4

Which test is most appropriate for determining the convergence of the series $$\sum_{n=1}^{\infty} \frac{n^n}{n!}$$?

Accepted Answer

Ratio Test

Question 5

Suppose $$\lim_{n 	o \infty} \left| \frac{a_{n+1}}{a_n} ight| = ho. $$According to the Ratio Test, what can be concluded if $$ho = 1$$?

Accepted Answer

The test is inconclusive.

Question 6

For the series $$\sum_{n=1}^{\infty} \frac{\cos n}{n^2$}$$, which test can be used to show absolute convergence?

Accepted Answer

Comparison Test

Question 7

Which of the following is NOT a requirement for the Alternating Series Test (Theorem 15)?

Accepted Answer

The terms $$u_n$$ are always negative.

Question 8

If $$\sum a_n$$ converges absolutely, what does Theorem 17 (Rearrangement Theorem) guarantee?

Accepted Answer

Any rearrangement of the series $$\sum a_n$$ converges absolutely and to the same sum.

Question 9

Which of the following series is divergent?

Accepted Answer

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

Question 10

A student claims that the series $$\sum_{n=1}^{\infty} \frac{(-1)^n}{n} is $$absolutely convergent. Which statement best explains why this is incorrect?

Accepted Answer

The series $$\sum_{n=1}^{\infty} \frac{1}{n}$$ diverges, so the original series is only conditionally convergent.

Question 11

Which statement about the harmonic series $$\sum_{n=1}^{\infty} \frac{1}{n} is $$correct?

Accepted Answer

It diverges.

Question 12

For the series $$\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$$, what is the value it converges to?

Accepted Answer

$$\ln 2$$

Question 13

Which test is most convenient to apply when nth powers occur in the terms of a series?

Accepted Answer

Root Test

Question 14

If the alternating series $$\sum_{n=1}^{\infty} (-1$$)^{n+1}$$ u_n$$ satisfies the three conditions of Theorem 15, what does Theorem 16 (Alternating Series Estimation Theorem) guarantee?

Accepted Answer

The error in approximating the sum by the nth partial sum is less than $$u_{n+1}.$$

Question 15

Which of the following is true about conditionally convergent series?

Accepted Answer

They converge, but not absolutely.

Question 16

A medical researcher is analyzing a series representing the probability of a rare event, modeled as $$\sum_{n=1}^{\infty} \frac{(-1$$)^{n+1}$$}{n!}. $$Which test should be used to estimate the sum and error?

Accepted Answer

Alternating Series Estimation Theorem

Question 17

Which of the following is a correct application of the Ratio Test for the series $$\sum_{n=1}^{\infty} \frac{n^3$}{3^n}$$?

Accepted Answer

The limit $$\lim_{n 	o \infty} \left| \frac{a_{n+1}}{a_n} ight| is $$less than 1, so the series converges absolutely.

Question 18

A chemical engineer is studying a series $$\sum_{n=1}^{\infty} \frac{1}{n^p}$$ for $$p \u003e 1. $$What can be concluded about the convergence of this series?

Accepted Answer

The series converges for all $$p \u003e 1.$$

Absolute and Conditional Convergence; The Ratio, Root, and Alternating Series Tests

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