2
Sequences and series
We will first deal with sequences, and then study infinite series in terms of
the associated sequence of partial sums.
2.1
Sequences
By a
sequence
, we will mean a collection of numbers
a
1
, a
2
, a
3
, . . . , a
n
, a
n
+1
, . . . ,
which is indexed by the set
N
of natural numbers. We will often denote it
simply as
{
a
n
}
.
A simple example to keep in mind is given by
a
n
=
1
n
, which appears to
decrease towards zero as
n
gets larger and larger. In this case we would like
to have 0 declared as the
limit of the sequence
. A quick example of a sequence
which does not tend to any limit is given by the sequence
{
1
,
−
1
,
1
,
−
1
, . . .
}
,
because it just oscillates between two values; for this sequence,
a
n
= (
−
1)
n
+1
,
which is certainly bounded.
Definition 2.1
A sequence
{
a
n
}
is said to converge, i.e., have a limit
A
,
iff for any
ε >
0
there exists
N
=
N
(
ε
)
>
0
s.t.
for all
n
≥
N
we have

a
n
−
A

< ε
. Notation:
lim
n
→∞
a
n
=
A
, or
a
n
→
A
as
n
→ ∞
.
Two Remarks
:
(i) It is immediate from the definition that for a sequence
{
a
n
}
to converge,
it is
necessary
that it be bounded.
However, it is
not suﬃcient
, i.e.,
{
a
n
}
could be bounded without being convergent. Indeed, look at the
example
a
n
= (
−
1)
n
+1
considered above.
(ii) It is only the
tail
of the sequence which matters for convergence. Oth
erwise put, we can throw away any number of the terms of the sequence
occurring at the beginning without upsetting whether or not the later
terms bunch up near a limit point.
Lemma 2.2
If
lim
n
→∞
a
n
=
A
and
lim
n
→∞
b
n
=
B
, then
1
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1)
lim
n
→∞
(
a
n
+
b
n
) =
A
+
B
2)
lim
n
→∞
(
ca
n
) =
cA
,
for any
c
∈
R
3)
lim
n
→∞
a
n
b
n
=
AB
4)
lim
n
→∞
a
n
/b
n
=
A/B
,
if
B
̸
= 0
.
Proof of 1)
For any
ε >
0, choose
N
1
and
N
2
so that for
n
1
≥
N
1
,
n
2
≥
N
2
,

a
n
1
−
A

< ε/
2,

b
n
−
B

< ε/
2. Then for
n
≥
max
{
N
1
, N
2
}
, we
have

a
n
+
b
n
−
A
−
B

<
ε
2
+
ε
2
=
ε.
Hence
A
+
B
is the limit of
a
n
+
b
n
as
n
→ ∞
.
2
Proofs of the remaining three assertions are similar.
For the last one,
note that since
{
b
n
}
converges to a nonzero number
B
, eventually all the
terms
b
n
will necessarily be nonzero, as they will be very close to
B
. So, in
the sequence
a
n
/b
n
, we will just throw away some of the initial terms when
b
n
= 0, which doesn’t affect the limit as the
b
n
occurring in the tail will all
be nonzero.
2
Example
:
We have lim
n
→∞
1
n
= 0. Indeed, given
ε >
0, we may choose
an
N
∈
N
such that
∃
N >
1
ε
, because
N
is unbounded. This implies that for
n
≥
N
, we have

1
n
−
0
 ≤
1
N
< ε
. Hence
{
1
n
}
converges to 0.
2
Definition
A sequence is said to be
monotone increasing
, denoted
a
n
↗
,
if
a
n
+1
≥
a
n
for all
n
≥
1, and
monotone decreasing
, denoted
a
n
↘
, if
a
n
+1
≤
a
n
for
n
≥
1. We say
{
a
n
}
is monotone (or monotonic) if it is of one
of these two types.
Theorem 2.3
A bounded, monotonic sequence converges.
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 Winter '08
 Borodin,A
 Sequences And Series, Candide, Mathematical analysis, Cauchy

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