Q: Which of the following statements is TRUE regarding the convergence of a sequence $$\{a_n\}$$?

If $$\lim\limits_{n \to \infty} a_n = L$$, then for every $$\epsilon \u003e 0$$, there exists an integer $$N$$ such that $$|a_n - L| N.$$

Q: Suppose $$\lim\limits_{x \to \infty} f(x) = L$$ and $$a_n = f(n)$$ for $$n \geq n_0$. What can be concluded?

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

Question 1

Which of the following best describes an infinite sequence of numbers?

Accepted Answer

A function whose domain is the set of positive integers.

Question 2

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

Accepted Answer

$$0$$

Question 3

Which of the following statements is TRUE regarding the convergence of a sequence $$\{a_n\}$$?

Accepted Answer

If $$\lim\limits_{n 	o \infty} a_n = L$$, then for every $$\epsilon \u003e 0$$, there exists an integer $$N$$ such that $$|a_n - L|  N.$$

Question 4

Which of the following sequences diverges to infinity?

Accepted Answer

$$a_n = n$$

Question 5

If $$\lim\limits_{n 	o \infty} a_n = 2$$ and $$\lim\limits_{n 	o \infty} b_n = 3$$, what is $$\lim\limits_{n 	o \infty} (a_n + b_n)$$?

Accepted Answer

$$5$$

Question 6

Suppose $$-\frac{1}{n} \leq a_n \leq \frac{1}{n}$$ for all $$n$$, and $$\lim\limits_{n 	o \infty} -\frac{1}{n} = 0$$ and $$\lim\limits_{n 	o \infty} \frac{1}{n} = 0. $$What is $$\lim\limits_{n 	o \infty} a_n by $$the Sandwich Theorem?

Accepted Answer

$$0$$

Question 7

Let $$a_n = \left(1 + \frac{x}{n}ight)^n$$ for any real number $$x. $$What is $$\lim\limits_{n 	o \infty} a_n$$?

Accepted Answer

$$e^x$$

Question 8

Which of the following is a necessary and sufficient condition for a nondecreasing sequence of real numbers to converge?

Accepted Answer

It is bounded from above.

Question 9

If $$a_n = x^n$$ and $$|x| \u003c 1$$, what is $$\lim\limits_{n 	o \infty} a_n$$?

Accepted Answer

$$0$$

Question 10

Suppose $$\{a_n\} is a $$sequence such that $$a_n \leq a_{n+1}$$ for all $$n$$ and $$a_n \leq 5$$ for all $$n. $$What can be said about the sequence?

Accepted Answer

It converges to a limit $$L \leq 5.$$

Question 11

Let $$a_n = \sqrt[n]{n}. $$What is $$\lim\limits_{n 	o \infty} a_n$$?

Accepted Answer

$$1$$

Question 12

If $$\lim\limits_{n 	o \infty} a_n = L$$ and $$f is $$continuous at $$L$$, what is $$\lim\limits_{n 	o \infty} f(a_n)$$?

Accepted Answer

$$f(L)$$

Question 13

Which of the following is NOT necessarily true for a bounded sequence $$\{a_n\}$$?

Accepted Answer

The sequence converges.

Question 14

Suppose $$\lim\limits_{x 	o \infty} f(x) = L$$ and $$a_n = f(n)$$ for $$n \geq n_0$. What can be concluded?

Accepted Answer

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

Question 15

If $$a_n = (-1)^n \frac{1}{n}$$, what is $$\lim\limits_{n 	o \infty} a_n$$?

Accepted Answer

$$0$$

Question 16

A sequence $$\{a_n\} is $$called nondecreasing if:

Accepted Answer

$$a_n \leq a_{n+1}$$ for all $$n$$

Question 17

If $$\lim\limits_{n 	o \infty} a_n = A$$ and $$\lim\limits_{n 	o \infty} b_n = B 
eq 0$$, what is $$\lim\limits_{n 	o \infty} \frac{a_n}{b_n}$$?

Accepted Answer

$$\frac{A}{B}$$

Question 18

Which of the following is an example of a sequence that diverges?

Accepted Answer

$$a_n = n$$

Question 19

If $$a_n = x$$^{n}$$ for $$|x| \u003e 1$$, what is $$\lim\limits_{n 	o \infty} a_n$$?

Accepted Answer

$$\infty$$

Sequences and Their Limits: Chapter 11.1 Study Notes

Guida di studio - Domande di pratica

Study guide - Flashcard