If $$a_n \leq b_n \leq c_n$$ for all $$n$$ and $$\lim_{n \to \infty} a_n = \lim_{n \to \infty} c_n = L$$, then $$\lim_{n \to \infty} b_n = L.$$

If $$\lim_{n \to \infty} a_n = A$$ and $$\lim_{n \to \infty} b_n = B$$, then $$\lim_{n \to \infty} (a_n b_n) = AB.$$

Question 1

Which of the following best describes an infinite sequence?

Accepted Answer

A list of numbers in a specific order, indexed by integers.

Question 2

Given the sequence $$a_n = \frac{1}{n}$$, what is $$\lim_{n 	o \infty} a_n$$?

Accepted Answer

$$0$$

Question 3

Which of the following is a correct application of the Squeeze (Sandwich) Theorem for sequences?

Accepted Answer

If $$a_n \leq b_n \leq c_n$$ for all $$n$$ and $$\lim_{n 	o \infty} a_n = \lim_{n 	o \infty} c_n = L$$, then $$\lim_{n 	o \infty} b_n = L.$$

Question 4

Suppose $$a_n = (-1)^n. $$Which of the following statements is true?

Accepted Answer

The sequence diverges.

Question 5

Which of the following sequences is both bounded and monotonic?

Accepted Answer

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

Question 6

If $$\lim_{n 	o \infty} a_n = A$$ and $$\lim_{n 	o \infty} b_n = B$$, what is $$\lim_{n 	o \infty} (a_n + b_n)$$?

Accepted Answer

$$A + B$$

Question 7

Given the recursive sequence $$a_1 = 2$, a$$_{n+1} = \frac{1}{2} a_n$$, what is a_3$?

Accepted Answer

$$0.5$$

Question 8

Which of the following is NOT a property of convergent sequences?

Accepted Answer

Every convergent sequence is monotonic.

Question 9

If $$a_n = \frac{n}{n+1}$$, what is $$\lim_{n 	o \infty} a_n$$?

Accepted Answer

$$1$$

Question 10

Which of the following statements about the sequence $$a_n = \frac{1}{n^2$} is $$true?

Accepted Answer

It is bounded and monotonic.

Question 11

Suppose $$a_n = \frac{1}{n}$$ and $$b_n = \frac{2}{n}. $$What is $$\lim_{n 	o \infty} (a_n \cdot b_n)$$?

Accepted Answer

$$0$$

Question 12

Which of the following is a correct recursive definition for the sequence $$a_n = 2^n$$?

Accepted Answer

$$a_1 = 2$$, a_{n+1} = 2a_n$$

Question 13

If $$a_n = \frac{1}{n}$$ and $$b_n = (-1)^n \frac{1}{n}$$, what can be said about the convergence of $$b_n$$?

Accepted Answer

$$b_n$$ converges to $$0.$$

Question 14

Which theorem guarantees that a bounded and monotonic sequence converges?

Accepted Answer

The Monotonic Sequence Theorem

Question 15

Given $$a_n = \frac{n}{n+2}$$, which of the following is true?

Accepted Answer

$$a_n is $$bounded and monotonic.

Question 16

Which of the following is a correct statement about the limit of a product of sequences?

Accepted Answer

If $$\lim_{n 	o \infty} a_n = A$$ and $$\lim_{n 	o \infty} b_n = B$$, then $$\lim_{n 	o \infty} (a_n b_n) = AB.$$

Question 17

Which of the following sequences is NOT monotonic?

Accepted Answer

$$a_n = (-1)^n$$

Question 18

If $$a_n = \frac{1}{n}$$ and $$b_n = 2$$, what is $$\lim_{n 	o \infty} (a_n + b_n)$$?

Accepted Answer

$$2$$

Question 19

Which of the following is a correct definition of a sequence?

Accepted Answer

A function whose domain is the set of positive integers.

Infinite Sequences and Their Limits

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