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Exercise 10.
1
0
1
0
0
2 2
1
2
0
2 2
1
n
n 1
1
1
n
n
n
k 0
n
2
0
2 4 6
0
1
0
2
!
0
cos
Hence,ifnisapositiveinteger,itfollows that
.
1!
1
!
0
cos x
have that
Then,ifxisarealnumbersatisfying 0 x 1 andifnisapositiveinteger, we
.
1!
1
R
Thus
f , 1.
Observe that
.
1!
,
R
between 0 andx such that
Formulathat thereisarealnumberc(n,x),dependingonbothnand x,
Forapositiveintegernandanonzerorealnumber x,wehavefromTaylor' s
R.
and let
,
!
0
S x
Forapositiveintegernandeachrealnumber x, let
2!
1
6!
1
4!
1
2!
1
1 -
!
0
TheTaylorseriesforf(x)about x 0 is
Letf(x) cos(x).
29.Consider cosx.
x dx
k
f
x dx
x
n
x
k
f
x
n
x
cn x
x
n
f cn x
x
x f x S x
x
k
f
j
x
x x x
x
n
f
dx
n
k
k
k
n
n
k
k
k
n
n
n
n
k
k
j
j
j
n
n
n
n 6 works.Thenourapproximation is
n 1! 2 3
beaccurateto 4 decimalplaces,choosenlargeenoughso that
cos x
to
Thus,inorder tohavetheapproximat ion
n 1! 2 3
n 1!
n 1!
cos x
cos x
1
0
5 9
1
0
4 8 12
1
0
2
1
0
0
2
1
0
2 3
2 2
1
0
1
0
0
2 2
1
0
0
2 2
x x x
x x x dx
n
dx
x dx
k
f
n
x
n
x dx
x dx
k
f
x dx
k
f
n
k
k
k
n
n
n
k
k
k
n
k
k
k
6
5 5
2 4
5
2
cos 5 3 3
cos() 3 2 3
cos() 4 2 2 2
cos() 2 2 2
cos() 3
cos() 3
cos( )
cos() 2
cos() 2
cos() cos
cos( )
cos( 0 ) 1
15 sin cos 15 sin cos sin.
sin 10 sin ( )cos( ) 10 sin ( )
6 cos( )sin( ) sin( )
cos 4 sin cos 12 sin cos 6 sin 8 sin cos
sin sin ( ) 6 sin ( )cos( ) 4 sin ( ) 3 cos ( )
sin 3 sin cos 3 cos( ) 3 sin ( ) cos().
( ) sin sin ( ) 3 sin()cos( )
sin 3 sin cos sin.
2 sin cos sin
sin sin ( ) cos( )
sin cos.
sin sin cos
( ) ( sin( )).
x
f c
R x f x S x
x
e
x
e
S x e
f
x x x x x
e x x x x
x x x
x e x x x x x x x
f x e x x x x x x
f e
x e x x x x x
f x e x x x x
f
e x x x x
e x x x
f x e x x x
f e
e x x
f x e x x e x
f
f x e x
f e e e
vi
v
x
x
v x
iv
x
iv x
x
x
x
x
x x
x
S x 1 ( ) ( ).
wehave that
So,letS(x)denotetheactualsumofthisseries.Foreachrealnumber xandeachpositiveinteger n,
However,wedonotknowfromthisthat thesumofthisseriesisactuallyf(x) e.
Assumingthispatterncontinues,thisserieswillconvergeforall x.
a ( )
a ,
where
TheTaylorseriesforf(x)about x 0 appearstobeanalternating series
15 cos( ) 15 cos ( ) cos( )
45 sin ( )cos ( ) 75 sin ( )cos( ) 16 sin ( )
sin 15 sin ( )cos( ) 20 sin ( )
30 sin cos 15 cos 15 sin ( ) cos
10 sin ( ) 30 sin ( )cos( ) 15 cos ( )
5 sin ( )cos( ) 30 sin ( )cos ( )
10 sin ( ) 15 sin( )cos ( ) 15 sin( )cos( ) sin( )
sin sin 10 sin ( ) cos
wherecisbetween 0 and x.
1
0
cos(x)
6
3
4
2
2
0 1
n 0
2 4 6
6
3 2
2 2 2 2
cos() 6 4 4
2 2 2
4 2 3
cos() 4 2 2
3 2
cos() 5 3
a x a x
x
e
x
x
e
x a x
e
x ea x
a x
x
e
x
e
x
e
S x e
f e
x x x
x x x x x
e x x x x
x x x x x
x x x x
e x x x x
x x x x x x
f x e x x x x
n
n
k
k
k
n
n
iv
x
x
vi x
Problem 33(e)
Taylorseriesforf(x) lncos ( )about x 0 :
( 0 ) lncos ( 0 ) ln( 1 ) 0.
tan( )
Taylorseriesfor tan(x)about x 0 :
2 cos( )sin( ) 2 tan( ).
cos ( )
( ) lncos ( )
4
2 4 6 8
2
2
2 4 6 8
3 5 7
2
2
f
x x x x
x
f
xdx x x x x c
x x x x
x x x
x
f x
f x x
4
f
