The Limit
Comparison
-
-
consider
an
f bn
suppose
where
Then
or
L
Ean
are
king
#
O
,
Ean &
Test
& { bn
positive
In
.
bn
both diverge
for all
n
,
=L
L too
E
,
where
.
both
converge
In
practice
given Ean
,
approximate
where
So
E
an
bn
Then work
[ bn
the
is
roughly
precisely
(c)
LCT
.
get
to
known
is
first work
to
or
,
we
bn
be (c)
&
to
if
apply
(D)
or
get E
to
to decide
(D)
,
b
.
.
Some useful approximations
For
small
=
sink
cos
X
x
oc
,
-
ate:
Iz
:
ln ( It )
X
K
txt
2
Itak
x
(
I
n
fantail
is
and
sink )
x
I
If
cost )
large th
t
K
2
:
hat 's )
( it lay
is
,
I
-
Ins
.
I
2
'
h
It
In
small
E.g
E
.
3n2t2ny
n3
t
I
32373ft
3¥
In
bunks
kiss (
an
=
a
=
=
31.int#Kgg--nEgg3n3t2n2tn3n3t3
=
=
this
3+2+423
7,3
t
=
Since
is
{ In
(D)
by
3-3
=3
the
=
{
*
I
is
8
(D) by
LC T
.
so
p series ,
-
E3n%t?nI
Eg
.
E
I
ans
him Inn
-
his
.
¥
=
In
{
Elon
-
is
is
n
large
.
Ez
¥a
.
I
=
( not
0
,
,
Since
bn if
-_
In
fig
=
-
=
Eth
CD )
-
IE 's
by
not
CD )
is
LCT
.
by
a)
p
-
series
,
E.
g
an
.
=
arc
tank )
=
.
=
=
so
dancer )
Euratom Cfa )
him
Since
's
x
fa
=
÷
En
-
=
-
-
Es
bn
"
so
kiss ¥¥a by
(E
I
E bn
(
=
not
o
E
Euratom (Tn )
not
a
,
is
is
Es)
(D)
by
%
if n
is large
"
L'Hop
)
by
p
-
series
,
LCT
.
.
E.g
{ Citi
.
Choice
Ca) :
(
'
v
.
an
Choice
=
l
1st
Use
tha )
"
't
( tha )
i
( b)
:
an
-
=
=
-
approx
I
I
1)
2
l
t
if
Ina
In
( Vith
:
=
-
bn
u
if
1) (Vith
large
is
n
.
)
ti
kt÷?x
if
large
is
=
n
Ina bn
-
is
large
.
tins
=
-
so
not
{ bn
so
0
=
,
¥
Ess
#¥+
kiss
not
,
Edith
¥
-
N
.
's Ent
-
=
D
is
is
Cc)
Cc )
by
by
p
-
series
LCT
.
I
E.g Ebing
.
could
But
tnf
.
.
.
{ &
is
bn =L
try
turn
is
(D)
but
>
>
I
if
Gn
for
(D) by
by
>
n
p
-
CT
.
LCT won't work
e
n -
series ,
e
so
E
hung
.
E.
g
[
.
hang
lnn
↳
>
a
CT won't
so
h
n
ri
if
e
work with ↳
for
any
sufficiently large
a
>
for
O
sorry
.
n
.
nhn-rn.hn#enIa--- ¥
:
.
.
If
u >
we
choose
work )
x
then
so
that
bn
=
n¥
E=
a
>
nst
I
(
and
x-
k, will
{¥
.
{
so
Eg
.
)
kung
=L (
Life
so
K
by
p
cc )
is
-
series
by
CT
.
Ek ( Itn )
an
=
is
I
it
ha)
hCYI
x
=
,
and
Ebell
is
cc )
b
if
n
.
is
large
.
L'
by Hop
kiss
Elon
since
tha )
La
=
C- Kid
is
(c) by
p
by LCT
.
-
series