# Sequences

### Example 1

Determine if the sequence is increasing, decreasing, or neither.

*a _{n}* = 2

^{n}### Example 2

Determine if the sequence is increasing, decreasing, or neither.

### Example 3

Determine if the sequence is increasing, decreasing, or neither.

*a _{n}* = (-1)

^{n}n^{2}

### Example 4

Determine if the sequence is increasing, decreasing, or neither.

*a _{n}* = -

*n*

^{2}

### Example 5

Determine if the sequence is increasing, decreasing, or neither.

### Example 6

Determine if the sequence is (a) bounded above, (b) bounded below, and (c) bounded or unbounded.

### Example 7

Determine if the sequence is (a) bounded above, (b) bounded below, and (c) bounded or unbounded.

*a _{n}* = 4 –

*n*

### Example 8

Determine if the sequence is (a) bounded above, (b) bounded below, and (c) bounded or unbounded.

*a _{n}* = (-1)

^{n}### Example 9

Determine if the sequence is (a) bounded above, (b) bounded below, and (c) bounded or unbounded.

*a _{n}* =

*n*

^{3}

### Example 10

Determine if the sequence is (a) bounded above, (b) bounded below, and (c) bounded or unbounded.

*a _{n}* = (-1)

*2*

^{n}

^{n}### Example 11

Determine if the statement is true or false. Explain your reasoning.

If a sequence has 5 ≤ *a _{n}* ≤ 6 for all

*n*, then the sequence must converge.

### Example 12

Determine if the statement is true or false. Explain your reasoning.

The sequence 1,1,1,1,... is both convergent and bounded.

### Example 13

Determine if the statement is true or false. Explain your reasoning.

If a sequence diverges, the sequence is unbounded.

### Example 14

Determine if the statement is true or false. Explain your reasoning.

If a sequence is unbounded, that sequence diverges.

### Example 15

Determine if the statement is true or false. Explain your reasoning.

If a sequence converges, there is some value *K* such that *K* ≤ *a _{n}* for all

*n*.