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 3.6.3If 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 3.6.4Let
be an arithmetic sequence whose difference is equal to
.
If
, the sequence is constant.
If
, then the sequence increases and has
as its limit.
If
, then the sequence increases and has
as its limit.
Theorem 3.6.5Let
be an geometric sequence whose quotient is equal to
.
If
, the sequence is constant.
If
, the sequence is constant from its second term.
If
, the sequence is convergent and its limit is 0.
If
, the sequence is divergent; its limit is infinite (
or
according to the sign of the first term).
If
, the sequence is divergent and has no limit.
If
, the sequence is periodic, with period equal to 2.