Exercises on Numerical Sequences

1

 

Is increasing

Is bounded below

Lower bounds:

The minimum is

Is not bounded above

Divergent

 

2

 

Is decreasing

Is bounded above

Upper bounds:

The maximum is

Is not bounded below

Divergent

 

3

 

Is decreasing

Is bounded above

Upper bounds:

The maximum is

Is bounded below

Lower bounds:

The infimum is

Convergent,

 

4

 

Is not monotonic

Is not bounded

Is neither convergent nor divergent

 

5

 

The first terms of this sequence are:

Is strictly monotonically decreasing

Convergent sequence

Since it is decreasing, is an upper bound, the maximum.

is a lower bound, the infimum or greatest lower bound.

Therefore the sequence is bounded

6

 

The first terms of the sequence are:

Is not monotonic

Is neither convergent nor divergent

Is not bounded

 

7

 

Is not monotonic

Is convergent because

Is bounded above, is the maximum

Is bounded below, is the minimum

Is bounded

 

8

 

The first terms of the sequence are:

Is strictly monotonically increasing

Convergent sequence

Is bounded below, is the minimum

Is bounded above. is the supremum

Therefore the sequence is bounded

1

 

We can obtain the difference between consecutive terms:

Since the difference is constant,

It is an arithmetic progression

 

2

 

We can divide each term by its predecessor:

Since the quotient is constant,

It is a geometric progression

 

3

 

The sequence can be rewritten as:

We observe that the bases are in arithmetic progression, with , and the exponent is constant, so we can write the following sequence for the base:

Therefore the general term is:

 

4

 

Each term of this sequence is one more than the terms of the previous sequence, so we can rewrite it as:

We find the general term as we saw in the previous case and add 1.

 

5

 

The sequence can be rewritten as:

 

 

6

 

The sequence can be rewritten as:

 

 

7

 

Each of the terms of this sequence is the opposite of each of the terms of sequence 3, so:

 

8

 

 

9

 

We have two sequences, one for the numerator and another for the denominator:

The first is an arithmetic progression with , the second is a sequence of perfect squares.

 

10

 

If we disregard the sign, the numerator is an arithmetic progression with .

The denominator is an arithmetic progression with .

Since the odd terms are negative, we multiply by .

1

 

The numerator is constant.

The denominator is an arithmetic progression with .

 

 

2

 

The numerator is an arithmetic progression with

The denominator is an arithmetic progression with

 

 

3

 

If we write each term of the sequence in rational form, we would obtain:

 

 

The numerator is an arithmetic progression with

The denominator is an arithmetic progression with

 

 

4

 

If we disregard the sign, it is an arithmetic progression with

Since the odd terms are negative, we multiply by

 

 

5

 

The sequence can be rewritten as:

If we disregard the sign, the numerator is an arithmetic progression with

The denominator is an arithmetic progression with

Since the even terms are negative, we multiply by

 

 

6

 

It is an oscillating sequence

The odd terms form an arithmetic progression with , if we don't take into account the even terms

The denominator of the even terms forms an arithmetic progression with

 

 

7

 

The sequence can be rewritten as:

If we disregard the sign and the exponent, we have an arithmetic progression with

Since the terms are squared, we have to square the general term

Since the odd terms are negative, we multiply by

 

 

8

 

The sequence can be rewritten as:

It is an oscillating sequence

The numerator of the odd terms forms an arithmetic progression with , if we don't take into account the even terms.

Since the terms are squared, we have to square the general term

The first addend of the denominator (disregarding the square) is an arithmetic progression with (not counting the even terms)

The general term must be squared and add

The even terms form a constant sequence.

 

 

9

 

Separating the sequences of the numerator and the denominator we have:

Numerator:

Denominator:

The numerator is an arithmetic progression with

The denominator is a geometric progression with

 

 

10

 

If we disregard the sign, the numerator is an arithmetic progression with

The denominator is a geometric progression with

Since the even terms are negative, we multiply by