Q: Let $$a_n = \frac{1}{n}. $$Which of the following statements is true?

$$\{a_n\}$$ converges to $$0 as$$ $$n \to \infty.$$

Q: Which of the following is a necessary condition for the convergence of the infinite series $$\sum\limits_{n=1}^{\infty} a_n$$?

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

Question 1

Which of the following best describes the definition of convergence for a sequence $$\{a_n\} to a $$limit $$L$$?

Accepted Answer

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

Question 2

Suppose $$\lim\limits_{n 	o \infty} a_n = 3$$ and $$\lim\limits_{n 	o \infty} b_n = 5. $$What is $$\lim\limits_{n 	o \infty} (2a_n - 3b_n)$$?

Accepted Answer

$$-9$$

Question 3

Which of the following sequences diverges to infinity?

Accepted Answer

$$a_n = n^2$

Question 4

Let $$a_n = \frac{1}{n}. $$Which of the following statements is true?

Accepted Answer

$$\{a_n\}$$ converges to $$0 as$$ $$n 	o \infty.$$

Question 5

Which of the following is a necessary condition for the convergence of the infinite series $$\sum\limits_{n=1}^{\infty} a_n$$?

Accepted Answer

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

Question 6

Consider the geometric series $$\sum\limits_{n=1}^{\infty} ar$$^{n-1}$$. For which values of r$ does the series converge?

Accepted Answer

$$|r| \u003c 1$$

Question 7

Suppose $$\{a_n\} is a $$bounded and nondecreasing sequence. What can you conclude about $$\{a_n\}$$?

Accepted Answer

$$\{a_n\}$$ converges.

Question 8

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

Accepted Answer

$$e$$

Question 9

Which of the following sequences is bounded but does NOT converge?

Accepted Answer

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

Question 10

Suppose $$\lim\limits_{n 	o \infty} a_n = L$$ and $$f is $$continuous at $$L. $$What is $$\lim\limits_{n 	o \infty} f(a_n)$$?

Accepted Answer

$$f(L)$$

Question 11

Let $$a_n = \frac{n!}{n^n}. $$What is $$\lim\limits_{n 	o \infty} a_n$$?

Accepted Answer

$$0$$

Question 12

Which of the following is true about the harmonic series $$\sum\limits_{n=1}^{\infty} \frac{1}{n}$$?

Accepted Answer

It diverges.

Question 13

If $$\sum\limits_{n=1}^{\infty} a_n$$ converges and $$\sum\limits_{n=1}^{\infty} b_n$$ diverges, what can you say about $$\sum\limits_{n=1}^{\infty} (a_n + b_n)$$?

Accepted Answer

It diverges.

Question 14

Which of the following is an example of a monotonic sequence?

Accepted Answer

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

Question 15

Suppose $$\lim\limits_{n 	o \infty} a_n = 0$$ and $$\lim\limits_{n 	o \infty} b_n = 0. $$Which theorem allows you to conclude $$\lim\limits_{n 	o \infty} (a_n + b_n) = 0$$?

Accepted Answer

The Sum Rule for limits of sequences

Question 16

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

Accepted Answer

$$1$$

Question 17

Which of the following statements about bounded sequences is true?

Accepted Answer

A bounded sequence may not converge.

Question 18

Suppose $$a_n = \frac{2}{n^2$}. $$What can you say about the series $$\sum\limits_{n=1}^{\infty} a_n$$?

Accepted Answer

It converges.

Question 19

If $$\sum\limits_{n=1}^{\infty} a_n$$ converges, what must be true about $$\lim\limits_{n 	o \infty} a_n$$?

Accepted Answer

$$0$$

Sequences and Infinite Series: Foundations and Properties

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