# Convergence and Divergence.

Definition 3.6.1   The sequence is convergent if there exists a real number verifying the following property (called Cauchy's inequality):

Otherwise, the sequence is divergent.

Notation: .

Example 3.6.2   Let ; we prove that is convergent and that

Take any ; we look for natural numbers such that . We have:

Let be the integer part of ; then for any natural such that , we have .

Proposition 3.6.3   If a sequence is convergent, then it is bounded.

Proof. Let be the limit of the sequence ; by definition 6.1 we have:

Thus the set is bounded (below by and above by ). As the set is finite, it is bounded. The set of all the terms of the sequence is the union of two bounded sets, hence it is bounded.

For example, any periodic sequence is bounded, but is divergent.

Theorem 3.6.4   Let be an arithmetic sequence whose difference is equal to .
1. If , the sequence is constant.
2. If , then the sequence increases and has as its limit.
3. If , then the sequence increases and has as its limit.

Theorem 3.6.5   Let be an geometric sequence whose quotient is equal to .
1. If , the sequence is constant.
2. If , the sequence is constant from its second term.
3. If , the sequence is convergent and its limit is 0.
4. If , the sequence is divergent; its limit is infinite ( or according to the sign of the first term).
5. If , the sequence is divergent and has no limit.
6. If , the sequence is periodic, with period equal to 2.

Definition 3.6.6   The sequence has as its limit if:

We denote: .

This formula is called Cauchy's inequality too. Write the corresponding definition for .

Noah Dana-Picard 2007-12-28