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Sequences 2
CHAPTER
2.4 Continuity
A sequence can be thought of as a list of numbers
written in a definite order: a1 ,a2 ,a3 ,a4 …,an ,…
The number a1 is called the first term, a2 is the
second term, and in general an is the nth term.
For every positive integer n there is a
corresponding number an and so a sequence can
be defined as a function whose domain is the set
of positive integers.
Notation: The
sequence { a1 ,a2 2
,a3 ,…} is also
CHAPTER
denoted by {an} or {an}n=1 .
2.4 Continuity
Definition: A sequence {an} has the limit L and
we write:
limn an = L or an L as n  
if we can make the terms an as close to L as we
like by taking n sufficiently large. If limn an
exists, we say the sequence converges (or is
convergent). Otherwise, we say the sequence
diverges (or is divergent).
Theorem: If CHAPTER
limn f(x) = L and2f(n)= an when n
is an integer, then limn an = L.
2.4 Continuity
Definition: A sequence {an} has the limit L and
we write:
limn an = L or an L as n  
if we can make the terms an as close to L as we
like by taking n sufficiently large. If limn an
exists, we say the sequence converges (or is
convergent). Otherwise, we say the sequence
diverges (or is divergent).
CHAPTER
2
If {an} and {bn} are convergent sequences and
2.4 Continuity
c is a constant,
then
1. limn (an + bn) = limn an + limn bn
2. limn (an - bn) = limn an - limn bn
3. limn can = c limn an
4. limn (an bn) = limn an . limn bn
5. limn (an / bn) = ( limn an / limn bn )
6. limn c = c
If an  bn CHAPTER
cn for n  n0 and 2
limn an = limn cn = L, then limn bn =
2.4 Continuity
L.
Theorem: If limn |an|= 0, then limn an = 0.
The sequence{rn}is convergent if –1 < r  1and
divergent for all other values of r.
limn rn = { 0 if –1 < r < 1
1 if r = 1
Definition: ACHAPTER
sequence {an}is called
2 increasing
if an < an+1 for all n  1, that is a1 < a2 < a3 <
2.4
Continuity
…. It is called decreasing if an > an+1 for all n
 1. A sequence monotonic if it’s either
increasing or decreasing.
Definition: ACHAPTER
sequence {an}is bounded
above if
2
there is a number M such that
2.4
Continuity
an  M for all n  1.
It is bounded below if there is a number m such
that
m  an for all n  1.
If it is bounded above and below, then {an}is a
bounded sequence.
Monotonic Sequence Theorem: Every bounded,
monotonic sequence is convergent.
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