CHAPTER 2
ALGEBRA
SERIES
,Convergence A a series
ndeterminant farms
-
L' Hopital's rate shows If tignafcxs stigma gcxs :
: =
=o or
*
In english :
then
stigma fgqg.tl?nafgIYkg if a fraction comes to to
them can take
you
the derivative at
-
the top and bottom .
Proof
f
##
f
2
tz
' "
+ (x a) (a) + ( x a) (a) +
fgl¥y
-
.
...
=
( Cast tz
' "
) + x a) g ( x a) 2g (a) +
- e. .
-
* laylar
series
fia )
§G[a#
f
[
tzcx
]
"
=
+ -
as (a) + e. .
g. ( al tzlx a) g ( as +
"
+ e.
-
-
.
BUT .
"
x =
a It
= fia )
* )
F#
fraction
*
NB check
always is
§ f- else
-
or or
L' Hop 's rule is invalid .
, tamp :
eim
x→0
arotanx
X
|/=aTotgI4=og
=
lim dadarctank ) By L' Hap
x→0
aaa ( k )
¥911
= lim
x→o )
=
1 .
Sinai
Example :
m
= end
1)
x→1 sinatx ) LI 0
||
=
→
o
=
,ydf,
Fahd L' Hap
By
aetxfsmlitx) ]
xttcasatx )
=
YY→1
= 1
(1) team
repeatedly if
*
you use L' Hop
may
= .
1- need be .
IT
ALGEBRA
SERIES
,Convergence A a series
ndeterminant farms
-
L' Hopital's rate shows If tignafcxs stigma gcxs :
: =
=o or
*
In english :
then
stigma fgqg.tl?nafgIYkg if a fraction comes to to
them can take
you
the derivative at
-
the top and bottom .
Proof
f
##
f
2
tz
' "
+ (x a) (a) + ( x a) (a) +
fgl¥y
-
.
...
=
( Cast tz
' "
) + x a) g ( x a) 2g (a) +
- e. .
-
* laylar
series
fia )
§G[a#
f
[
tzcx
]
"
=
+ -
as (a) + e. .
g. ( al tzlx a) g ( as +
"
+ e.
-
-
.
BUT .
"
x =
a It
= fia )
* )
F#
fraction
*
NB check
always is
§ f- else
-
or or
L' Hop 's rule is invalid .
, tamp :
eim
x→0
arotanx
X
|/=aTotgI4=og
=
lim dadarctank ) By L' Hap
x→0
aaa ( k )
¥911
= lim
x→o )
=
1 .
Sinai
Example :
m
= end
1)
x→1 sinatx ) LI 0
||
=
→
o
=
,ydf,
Fahd L' Hap
By
aetxfsmlitx) ]
xttcasatx )
=
YY→1
= 1
(1) team
repeatedly if
*
you use L' Hop
may
= .
1- need be .
IT
