Calculus
Math 21C, Fall 2010
Summary of basic results about sequences and series.
1. Sequences
Definition of limit.
A sequence
{
a
n
}
converges to a limit
L
as
n
→ ∞
,
written
lim
n
→∞
a
n
=
L,
or
a
n
→
L
as
n
→ ∞
if for every
ϵ >
0 there exists a number
N
such that

a
n

L

< ϵ
for all
n > N.
A sequence that does not converge is said to diverge.
Diverges to infinity.
A sequence
{
a
n
}
diverges to
∞
, written
lim
n
→∞
a
n
=
∞
,
if for every number
M
there exists
N
such that
a
n
> M
for all
n > N
Note that such a sequence does not have a finite limit
L
, so it is divergent
not convergent.
Upper bound criterion.
If
{
a
n
}
is an increasing sequence (
a
n
+1
≥
a
n
)
that is bounded from above (there exists a number
M
such that
a
n
≤
M
for
every
n
) then
{
a
n
}
converges.
Sandwich theorem.
If
a
n
≤
b
n
≤
c
n
and the limits
lim
n
→∞
a
n
=
L,
lim
n
→∞
c
n
=
L
exist and are equal, then
lim
n
→∞
b
n
=
L.
Continuous functions of sequences.
If lim
n
→∞
a
n
=
L
and
f
is a function
that is defined on some open interval containing
L
and is continuous at
L
,
then
lim
n
→∞
f
(
a
n
) =
f
(
L
)
.
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 Fall '01
 NA
 Geometric Series, Mathematical Series, converges

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