University Mathematics Handbook (2015)

IV. Single-Variable Differential Calculus

Chapter 1. Sequence, Limit of a Sequence

1.1  Definitions

A real numbers sequence is function .

is called the -th member of sequence, or general term formula.

In other words, a sequence is an infinite ordered set of real numbers , written in short as .

Sequences and are equal if for all .

Examples:

a.  Arithmetic sequence is a sequence where each term, starting from the second one, equals the previous term plus a constant number called the common difference.

is its general term formula, where is the first member.

is the -th partial sum of the arithmetic sequence.

b.  A geometric sequence is a sequence where each term, starting from the second one, equals the preceding term multiplied by a constant number called the common ratio of the sequence.

is its general term formula.

is the -th partial sum of the sequence.

c.  If, in an infinite geometric sequence, , it has a finite sum:

then the sum on the left side is called converging geometric series (see VI, 1.1).

d.  Sequence is called harmonic sequence, and is the general term formula.

e.  A Leibniz sequence is , where is the general term formula.

f.  In a sequence such as , consisting of one real number recurring infinitely, is the general term.

1.2  Recursive Definition of a Sequence

a.  The recursive definition of sequence has two parts:

1.  The first term or first terms value.

2.  Recursion formula when function is defined for all .

b.  Example

A given sequence is defined as:

.

To find , let's substitute , and the result will be .

Using to calculate , we obtain

.

1.3  Bounded Sequences

a.  Sequence is bounded above if there exists number , called upper bound, such that for all natural .

b.  The smallest upper bound of a sequence is called the sequence supremum, and is denoted as . If is a term of the sequence, it is called maximum, and is denoted as .

c.  Sequence is called bounded from below if there exists number , called lower bound, such that for all .

d.  Greatest lower bound of a sequence is called infimum, and is denoted as . If it is a member of the sequence, it is called minimum and denoted as .

e.  Sequence is bounded if it is bounded from above and below. In the same manner, sequence is bounded if there exists number such that for all natural there holds .

1.4  Increasing and Decreasing Sequences

a.  Sequence is increasing if there exists such that for all , .

b.  A sequence is strictly increasing if, for all , .

c.  A sequence is decreasing if, for all , .

d.  A sequence is strictly decreasing if, for all , .

e.  An increasing/decreasing, or strictly increasing/decreasing, sequence is called a monotonic sequence.

Example: is a strictly increasing sequence.

Proof:

From

it follows that for all natural .

1.5  Sub-Sequence

Sequence is a sub-sequence of sequence if all its terms are in sequence by the same order in which they appear .

Examples:

a.  Sequence is a sub-sequence of sequence .

b.  Sequence is a sub-sequence of sequence .

1.6  Limit of a Sequence

is the limit of sequence if, for every there exists natural number such that, for all , there holds or . In other words: for every given , there is only a finite number of terms which are not in interval . It is written or .

If a sequence has a limit, it is called convergent.

If a series has no limit, it is called divergent.

1.7  Properties of Convergent Sequences

a.  If a sequence has a limit, it is unique.

b.  If a sequence converges to some limit, then every sub- sequence of it converges to the same limit.

c.  If a sequence has two sub-sequences converging to two different limits, then the sequence diverges.

Example: The sequence diverges because it has two sub-sequences converging to different limits:

If , then sequence converges to zero.

If , then sequence converges to .

d.  Any converging sequence is bounded. The reverse proposition is incorrect.

Example: is bounded but not convergent sequence.

e.  If sequences and converge, then:

1.  For every constant ,

2.  

3.  

4.  If and also, then .

f.  If sequence is bounded and , then .

Example: The sequence converges to zero, and sequence is bounded by , therefore, .

g.  Squeeze (sandwich) theorem: if , , , are three sequences where holds and for all starting from , then, .

h.  If the sequence converges to , and starting from , , then .

i.  Cauchy's test: real numbers sequence converges if, and only if,, for every there exists a natural number such that for all there holds .

1.8  Examples of Limits

a.  

b.  

c.  Let be a sequence of positive numbers.

If there exists a limit , then the sequence converges and .

d.  If the sequence converges and , then:

1.  

2.  

3.  

4.  

e.  If

are two polynomials, and suppose that for all , then

.

1.9  Convergence to Infinity

a.  Sequence tends to infinity or minus infinity, if for every real number there only exists a finite number of terms smaller or greater than . In other words, for every there exists such that for all holds or .

It is written or .

b.  A sequence converges in the broad sense if it converges in the ordinary sense or it tends to infinity or minus infinity.

1.10  Limits of Monotonic Sequences. The Number

a.  Every monotonic and bounded sequence, either increasing or decreasing, converges. Every monotonic sequence converges in the broad sense.

b.  Every increasing, unbounded sequence tends to infinity.

c.  Every decreasing, unbounded sequence tends to negative infinity.

d.  The sequence is increasing and bounded, and and therefore, it had a finite limit denoted as

1.11  Cantor Theorem

Let be a sequence of closed intervals such that, for all , holds

, .

Then, there is only one point belonging to all intervals .

1.12  Bolzano-Weierstrass Theorem

Every bounded sequence has a converging sub-sequence.