SEQUENCES
A
sequence
{
a
n
}
∞
n
=1
is an ordered list of numbers. Thinks of as
{
a
n
}
∞
n
=1
=
{
a
1
, a
2
, a
3
, a
4
, . . .
}
goes on forever
⇑
.
You can also think of a sequence as a function
f
:
N
→
R
with
f
(
n
) =
a
n
.
Def 11.1.1: limit of a sequence (intuition).
A sequence
{
a
n
}
has the
limit
L
and we write
lim
n
→∞
a
n
=
L
or
a
n
→
L
as
n
→ ∞
if we can make the terms
a
n
as close to
L
as we like by taking
n
sufficiently large. If lim
n
→∞
a
n
exists, we say the
sequence
converges
(or is
convergent
). Otherwise we say the sequence
diverges
(or is
divergent
).
Def 11.1.2: limit of a sequence (precise).
A sequence
{
a
n
}
has the
limit
L
and we write
lim
n
→∞
a
n
=
L
or
a
n
→
L
as
n
→ ∞
if for each
ε >
0 there is a corresponding integer
N
such that if
n > N
then

a
n

L

< ε
. In shorthand
∀
ε >
0
∃
N
∈
N
such that
[
n > N
=
⇒ 
a
n

L

< ε
]
or equivalently
∀
ε >
0
∃
N
∈
N
such that
[
n > N
=
⇒
L

ε < a
n
< L
+
ε
]
Fun
Compare Def 11.1.2 (page 677, limit of a sequence) to Def 2.6.7 (page 138, limit of a function).
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 Fall '11
 KUSTIN
 Calculus, Limit, lim, Limit of a sequence

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