# Sequences Problems

### Solutions

#### Interpret the Following Sequences and Describe Them.

1**a _{n} = 1, 2, 3, 4, 5, ... n**.

2**a _{n} = -1, −2,−3, −4, −5, ... −n**.

3**a _{n} = 2, 3/2, 4/3, 5/4, ..., n+1/n. **

4**a _{n}= 2, −4, 8, −16, 32, ..., (−1)^{n−1} 2^{n}**.

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## 1

**a _{n} = 1, 2, 3, 4, 5, ... n**

It is increasing.

It is bounded below.

1 is the infimum.

It is not bounded above.

It is divergent.

## 2

**a _{n} = −1, −2,−3, −4, −5, ... −n**

It is decreasing.

It is bounded above.

−1 is the supremum.

It is not bounded below.

It is divergent.

## 3

**a _{n} = 2, 3/2, 4/3, 5/4, ..., n+1/n **

It is decreasing.

It is bounded above

2 is the supremum.

It is bounded below.

1 is the infimum.

It is convergent and the limit is 1.

## 4

**a _{n}= 2, −4, 8, −16, 32, ..., (−1)^{n−1} 2^{n}**

It is not monotonous

It is not bounded.

It is neither convergent nor divergent.

## 5

#### Monotone

3, 4/3, 1, 6/7,...

It is strictly monotonically decreasing.

#### Limit

a_{1}= 3

a_{3}= 1

a_{1 000}= 0.5012506253127

a_{1 000 000 }= 0.5000012500006

The limit is 0.5.

It is a convergent sequence.

#### Bounded

As the sequence is decreasing, 3 is an upper bound and the supremum.

0.5 is a lower bound and the infimum.

Thus, the sequence is bounded.

**0.5 < a _{ n }≤ 3**.

## 6

2, − 4, 8, −16, ...

**It is not monotonous.**

It is neither convergent nor divergent.

It is not bounded.

## 7

**It is not monotonous.**

It is convergent, the limit is 0.

It is bounded above and 1 is the supremum.

It is bounded below and −1 is the infimum.

It is bounded.

−1 ≤a_{ n } ≤ 1.

## 8

#### Monotone

It is strictly monotonically increasing.

#### Limit

a_{1}= 0.5.

a_{3}= 0.6666 .

a_{1000}= 0.999000999001.

a_{1000 000 }= 0.999999000001.

The limit is 1.

It is a convergent sequence.

#### Bounded

It is bounded below and 1/2 is the infimum.

It is bounded above and 1 supremum.

The sequence is bounded.

0.5 ≤ a_{ n }< 1