Sequences

#todo growing, shrinking and monotone sequence (monotone is both growing and shrinking)

Bounding

The sequence can be:

Upper bounded if such that for all .
Lower bounded if such that for all .
If a sequence meets both of these conditions, it is simply called bounded.

In some cases, the bounds can also serve as sequence maximums and minimums:

A lower bound can be referred to as a minimum if it is the largest lower bound and is a part of the sequence .
Similarly, an upper bound can be called a maximum if it is the smallest upper bound and belongs to the sequence $a_n.

For example, let's consider the sequence defined by .

This sequence is upper bounded by 1, and 1 serves as the sequence's maximum.
It is also lower bounded by 0, although the lower bound is not the sequence minimum since it is not part of the sequence.

Convergence

We can define a sequence as converging to a certain value () when for any epsilon value selected that is above 0 () exists a term in the sequence () such that .
#todo add as mathematical expression

Divergence

We can define a sequence as diverging ( or ) if:
#todo describe in text
We can write:
A sequence is said to be oscillating if it is neither convergent or divergent.

If a sequence is convergent then it is also bounded.
If a sequence is divergent () then it is not upper bounded (lower bounded).

Limit arithmetic

Undetermined values

Properties of limits with sequences

If and
()

Stoltz theorem

If we have a and:

and is shrinking with
or
is growing and

Square root principle

If a is not determined (), but exists, then also exists.
Ex Sequence exercises > Square root priniciple

Sandwich principle

If we have
and
Then exists
Ex Sequences > Sandwich principle

Subsequences

A subsequence of a sequence is a sequence of terms of where each next selected index (k) in the subsequence is bigger than the last (i.e. ).

Subsequences

A subsequence of a sequence is essentially a new sequence, denoted as , which is formed by picking terms from the original sequence , is always greater than the index of the previous term (i.e., ).

Note

The actual values of the selected terms don't impact whether a sequence qualifies as a subsequence of another sequence. What matters is the pattern of index growth, as expressed in the inequality