Q: Which of the following is a necessary condition for the convergence of the series $$\sum a_n$$?

$$\lim\limits_{n \to \infty} a_n = 0$$

Q: Which of the following is a correct application of the Ratio Test?

If $$\lim\limits_{n \to \infty} \left| \dfrac{a_{n+1}}{a_n} \right| = 0.5$$, the series converges.

Question 1

Which of the following best describes the sequence $$\left\{ a_n ight\}$$ where $$a_n = \dfrac{2n+1}{1-3\sqrt{n}} as$$ $$n 	o \infty$$?

Accepted Answer

The sequence diverges to $$-\infty.$$

Question 2

Which of the following is the correct definition of convergence for a sequence $$\{a_n\}$$?

Accepted Answer

For every positive number $$\epsilon$$, there exists an integer $$N$$ such that for all $$n \u003e N$$, $$|a_n - L| \u003c \epsilon.$$

Question 3

Given the sequence $$a_n = \cos(n\pi)/5^n$$, what is the sum $$\sum_{n=0}^{\infty} a_n$$?

Accepted Answer

$$\dfrac{6}{24}$$

Question 4

Which of the following series converges by the $$p-$$series test?

Accepted Answer

$$\sum\limits_{n=1}^{\infty} \dfrac{1}{n^2$}$$

Question 5

Which of the following statements about the geometric series $$\sum\limits_{n=1}^{\infty} ar$$^{n-1}$$ is true?

Accepted Answer

It converges if $$|r| \u003c 1$$ and the sum is $$\dfrac{a}{1-r}.$$

Question 6

Consider the series $$\sum\limits_{n=1}^{\infty} \dfrac{1}{n(\ln n)^2$}. $$Which test is most appropriate to determine its convergence?

Accepted Answer

Integral Test

Question 7

Which of the following is a necessary condition for the convergence of the series $$\sum a_n$$?

Accepted Answer

$$\lim\limits_{n 	o \infty} a_n = 0$$

Question 8

Which of the following series is absolutely convergent but not conditionally convergent?

Accepted Answer

$$\sum\limits_{n=1}^{\infty} \dfrac{1}{n^2$}$$

Question 9

Suppose $$\sum a_n$$ converges and $$\sum b_n$$ diverges. What can you say about $$\sum (a_n + b_n)$$?

Accepted Answer

It diverges.

Question 10

Which of the following is true about the sequence $$a_n = \dfrac{\ln n}{n} as$$ $$n 	o \infty$$?

Accepted Answer

$$a_n 	o 0$$

Question 11

Which theorem allows you to conclude that if $$a_n \leq b_n \leq c_n$$ for all $$n \u003e N$$ and $$\lim\limits_{n 	o \infty} a_n = \lim\limits_{n 	o \infty} c_n = L$$, then $$\lim\limits_{n 	o \infty} b_n = L$$?

Accepted Answer

Sandwich Theorem

Question 12

Which of the following series diverges by the $$nth-$$term test for divergence?

Accepted Answer

$$\sum\limits_{n=1}^{\infty} \dfrac{1}{n}$$

Question 13

Which of the following is a correct application of the Ratio Test?

Accepted Answer

If $$\lim\limits_{n 	o \infty} \left| \dfrac{a_{n+1}}{a_n} ight| = 0.5$$, the series converges.

Question 14

Which of the following is true about a sequence that is both bounded and monotonic?

Accepted Answer

It converges.

Question 15

Which of the following is the sum of the geometric series $$\sum\limits_{n=1}^{\infty} \dfrac{1}{9} \left(\dfrac{1}{3}\$$right)^{n-1}$$?

Accepted Answer

$$\dfrac{1}{6}$$

Question 16

Which of the following is a correct statement about the alternating series $$\sum\limits_{n=1}^{\infty} (-1$$)^{n+1}$$ \dfrac{1}{n}$$?

Accepted Answer

It converges conditionally.

Question 17

Which of the following is the correct application of the Comparison Test?

Accepted Answer

If $$0 \leq a_n \leq b_n$$ and $$\sum b_n$$ converges, then $$\sum a_n$$ converges.

Question 18

Which of the following is true about the sequence $$a_n = \dfrac{n}{n^2$ + 3} as$$ $$n 	o \infty$$?

Accepted Answer

$$a_n 	o 0$$

Question 19

Which of the following is the correct statement about the series $$\sum\limits_{n=1}^{\infty} \dfrac{5}{n+1}$$?

Accepted Answer

It diverges because it is a harmonic series.

Infinite Sequences and Series: Convergence, Divergence, and Tests

Study Guide - Practice Questions

Study Guide - Flashcards