$$a_n \u003c a_{n+1}$$ \text{ for all } n \geq 1$

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

Question 1

Which of the following is the precise definition of the limit of a sequence $$\{a_n\}$$ converging to $$L$$?

Accepted Answer

For every $$\epsilon \u003e 0$$, there exists an integer $$N$$ such that if $$n \u003e N$$, then $$|a_n - L| \u003c \epsilon.$$

Question 2

Given the sequence $$a_n = \frac{n}{n+1}$$, determine its limit as $$n 	o \infty.$$

Accepted Answer

$$1$$

Question 3

Which of the following statements is TRUE about bounded monotonic sequences?

Accepted Answer

Every bounded monotonic sequence is convergent.

Question 4

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

Accepted Answer

$$0$$

Question 5

Let $$a_n = \ln(n)/n. $$What is $$\lim_{n 	o \infty} a_n$$?

Accepted Answer

$$0$$

Question 6

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

Accepted Answer

$$a_n \u003c a_{n+1}$$ 	ext{ for all } n \geq 1$

Question 7

Determine if the sequence $$a_n = (-1)^n$$ converges.

Accepted Answer

No, it diverges.

Question 8

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

Accepted Answer

$$f(L)$$

Question 9

Which of the following sequences is bounded above and below?

Accepted Answer

$$a_n = \sin(n)$$

Question 10

Let $$a_n = \frac{n^2$}{n^2$+1}. Is $$the sequence increasing, decreasing, or monotonic?

Accepted Answer

Increasing and monotonic

Question 11

Find the bounds for the sequence $$a_n = \frac{1}{n}.$$

Accepted Answer

Bounded below by $$0$$ and above by $$1$$

Question 12

Suppose $$a_n = \frac{n}{10}. $$What is $$\lim_{n 	o \infty} a_n$$?

Accepted Answer

$$\infty$$

Question 13

Which of the following is an example of a sequence that is bounded but not monotonic?

Accepted Answer

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

Question 14

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

Accepted Answer

$$0$$

Question 15

Let $$a_n = \frac{n^2$}{n^2$+1}. $$What is $$\lim_{n 	o \infty} a_n$$?

Accepted Answer

$$1$$

Question 16

Which of the following is the correct application of the Squeeze Theorem for sequences?

Accepted Answer

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

Question 17

A sequence $$\{a_n\} is $$called bounded above if:

Accepted Answer

There exists $$M$$ such that $$a_n \leq M$$ for all $$n \geq 1$$

Question 18

Consider the sequence $$a_n = \frac{1}{n^2$}. Is it $$bounded and does it converge?

Accepted Answer

It is bounded and converges to $$0$$

Question 19

Let $$a_n = 1 - \frac{1}{n}. $$What is $$\lim_{n 	o \infty} a_n$$?

Accepted Answer

$$1$$

Limits and Properties of Sequences

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