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N A S T A V A M A T E M A T I K E N A F A K U L T E T I M A
D r B r a n k o S a v i
J E D A N P O S T U P A K R A Z V I J A ; A F U N K C I J E
U
S T E P E N I R E D . S U M I R A ; E S T E P E N I H R E D O V
A
U r a d u 5 ] u v e d e n j e j e d a n p o s t u p a k r a z v
i j a a f u n k c i j a u s t e p e n i r e d . C i
o v o g r a d a j e d a , n a o s n o v u p o s t u p k a r a z
v i j a a r a c i o n a l n e f u n k c i j e u s t e p e n i
r e d , u v e d e p o j a m p r o x i r e n o g z b i r a s t e
p e n o g r e d a i b r o j n o g r e d a k o j i i z o v o g a
n a s t a j e z a k o n k r e t n u v r e d n o s t p r o m e n
i v e .
1 . O p x t a f o r m u l a z a o d r e i v a e k o e f i c i j
e n a t a s t e p e n o g r e d a
F u n k c i j a f ( x ) a n a l i t i q k a u t a q k i x = 0 m
o e s e p r e d s t a v i t i s t e p e n i m
r e d o m
( 1 ) f ( x ) =
1
P
i = 1
a
i
x
i
g d e s u k o e f i c i j e n t i a
i
j e d n o z n a q n o o d r e e n i f u n k c i j o m f ( x ) p
o m o u f o r m u l a
( 2 ) a
i
=
f
( i )
( 0 )
i !
i = 0 1 . . . :
I z v e x e m o j e d n u o p x t u f o r m u l u z a o d r e i
v a e k o e f i c i j e n a t a s t e p e n o g r e d a ( 1 )
k o j a k a o p o s e b a n s l u q a j o b u h v a t a f o r m
u l u ( 2 ) .
T e o r e m a . A k o s e f u n k c i j a f ( x ) m o e p r e d
s t a v i t i s t e p e n i m r e d o m
( 1 ) , t a d a z a k o e f i c i j e n t e a
i
( i = 0 1 . . . ) v a i f o r m u l a
( 3 ) a
i
=
( i ; k ) !
i !
l i m
x ! 0
1
x
i ; k
d
k
R
i ; 1
( x )
d x
k
i = 0 1 . . .
g d e j e R
i ; 1
( x ) = f ( x ) ;
i ; 1
P
k = 0
a
k
x
k
, a k j e m a k o j i c e o b r o j k o j i z a d o v o a v a u
s l o v
0 6 k 6 i .
D o k a z . A k o s e k p u t a ( 0 6 k 6 i ) p r i m e n i L o
p i t a l o v o p r a v i l o n a d e s n u
s t r a n u j e d n a k o s t i a
i
= l i m
x ! 0
R
i ; 1
( x )
x
i
, d o b i e s e
a
i
=
1
i ( i ; 1 ) ( i ; k + 1 )
l i m
x ! 0
1
x
i ; k
d
k
R
i ; 1
( x )
d x
k
x t o j e n a d r u g i n a q i n z a p i s a n a f o r m u l a
( 3 ) . O v i m j e t e o r e m a d o k a z a n a .
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R a z v i j a e f u n k c i j e u s t e p e n i r e d 4 1
P o s l e d i c a 1 . Z a n a j v e e k , t j . z a k = i , i z
( 3 ) s e d o b i j a f o r m u l a
a
i
=
1
i
l i m
x ! 0
f
( i )
( x ) =
f
( i )
( 0 )
i !
:
P o s l e d i c a 2 . Z a n a j m a i k , t j . z a k = 0 , i z
( 3 ) s e d o b i j a f o r m u l a
( 4 ) a
i
= l i m
x ! 0
f ( x ) ; S
i ; 1
( x )
x
i
S
i ; 1
( x ) =
i ; 1
P
k = 0
a
k
x
k
:
A k o j e f u n k c i j a f ( x ) , a n a l i t i q k a u t a q
k i x = 0 , r a c i o n a l n a ,
( 5 ) f ( x ) =
A
0
+ A
1
x + A
2
x
2
+ + A
r
x
r
B
0
+ B
1
x + B
2
x
2
+ + B
s
x
s
z a k o e f i c i j e n t e a
i
( i = 0 1 . . . ) u ( 4 ) v r e d e f o r m u l e
( 6 ) a
0
=
A
0
B
0
a
i
=
1
B
0
A
i
;
i
P
k = 1
B
k
a
i ; k
B
0
6= 0 i = 1 2 . . . :
Z a i s t a , f o r m u l a ( 4 ) n a o s n o v u ( 5 ) p o s t
a j e
a
i
= l i m
x ! 0
A
0
+ A
1
x + + A
r
x
r
; ( B
0
+ B
1
x + + B
s
x
s
) S
i ; 1
( x )
( B
0
+ B
1
x + + B
s
x
s
) x
i
p a a k o s e i p u t a p r i m e n i L o p i t a l o v o p r a
v i l o n a d e s n u s t r a n u o v e j e d n a k o s t i ,
d o b i e s e f o r m u l a ( 6 ) .
N a p o m e n a 1 . U r a d u 5 ] i z v e d e n a j e j e d n a
o p x t i j a f o r m u l a .
P r i m e r 1 . F u n k c i j u f ( x ) =
1 2 ; 5 x
6 ; 5 x ; x
2
r a z v i t i u s t e p e n i r e d .
R e x e e . N a o s n o v u f o r m u l e ( 6 ) j e
a
0
= 2 a
1
=
5
6
a
2
=
3 7
6
2
. . . a
i
= 1 +
( ; 1 )
i
6
i
. . .
p a j e
1 2 ; 5 x
6 ; 5 x ; x
2
=
1
P
i = 0
h
1 +
( ; 1 )
i
6
i
i
x
i
:
P r i m e r 2 . F u n k c i j u f ( x ) = l n ( 1 + 3 x + 2
x
2
) r a z v i t i u s t e p e n i r e d .
R e x e e . N a o s n o v u f o r m u l e ( 6 ) j e
f
0
( x ) =
3 + 4 x
1 + 3 x + 2 x
2
=
1
P
i = 0
( ; 1 )
i
( 1 + 2
i + 1
) ] x
i
i d a e
Z
f
0
( x ) d x =
Z
1
P
i = 0
( ; 1 )
i
( 1 + 2
i + 1
) ] x
i
d x
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4 2 B . S a v i
o d n o s n o
l n ( 1 + 3 x + 2 x
2
) =
1
P
i = 0
( ; 1 )
i
1 + 2
i + 1
i + 1
x
i + 1
+ C :
K a k o j e f ( 0 ) = 0 , t o j e C = 0 , p a j e l n ( 1 + 3 x
+ 2 x
2
) =
1
P
i = 0
( ; 1 )
i
1 + 2
i + 1
i + 1
x
i + 1
.
N a p o m e n a 2 . F o r m u l a ( 6 ) s e m o e p r i m e n i
t i n e s a m o k a d a s u u ( 5 ) p o l i -
n o m i m a k o g s t e p e n a ( s t e p e n i s e e k s p l i
c i t n o i n e p o j a v u j u u f o r m u l i ( 6 ) ) , v e
i k a d a s e u m e s t o o v i h p o l i n o m a n a l a z e s
t e p e n i r e d o v i ( j e d a n i l i o b a ) . T a k o s e
,
n a p r i m e r , z a f u n k c i j u
t g x =
s i n x
c o s x
=
x ;
x
3
3 !
+
x
5
5 !
;
1 ;
x
2
2 !
+
x
4
4 !
;
=
0 + x + 0 x
2
;
x
3
3 !
+ 0 x
4
+
1 + 0 x ;
x
2
2 !
+ 0 x
3
+
x
4
4 !
+
n a o s n o v u f o r m u l e ( 6 ) d o b i j a
t g x = x +
x
3
3 !
+
2 x
5
1 5
+
1 7 x
7
3 1 5
+ :
P r i m e r 3 . F u n k c i j a f ( x ) =
x
e
x
; 1
d e f i n i s a n a j e z a s v a k o x o s i m z a x = 0 .
K a k o j e l i m
x ! 0
f ( x ) = 1 , u z i m a s e d a j e f ( 0 ) = 1 , p a a k o s e
f u n k c i j a e
x
z a m e n i
M a k l o r e n o v i m r e d o m , d o b i j a s e
f ( x ) =
1
1 +
x
2 !
+
x
2
3 !
+
x
3
4 !
+
a o d a v d e , n a o s n o v u f o r m u l e ( 6 ) ,
( a ) f ( x ) = 1 ;
1
2
x
1 !
+
1
6
x
2
2 !
;
1
3 0
x
4
4 !
+ :
N a o s n o v u f o r m u l e ( 2 ) b i s m o i m a l i
( b ) f ( x ) = f ( 0 ) +
x
1 !
f
0
( 0 ) +
x
2
2 !
f
0 0
( 0 ) +
x
3
3 !
f
0 0 0
( 0 ) + :
B r o j e v i f ( 0 ) = B
0
, f
0
( 0 ) = B
1
, f
0 0
( 0 ) = B
2
, f
0 0 0
( 0 ) = B
3
, . . . , s u B e r n u l i j e v i
b r o j e v i . U p o r e i v a e m ( a ) i ( b ) d o b i j a s
e :
B
0
= 1 B
1
= ;
1
2
B
2
=
1
6
B
3
= 0 B
4
= ;
1
3 0
B
5
= 0 B
6
=
1
4 2
. . .
I z r a q u n a v a e B e r n u l i j e v i h b r o j e v a p o
m o u i z v o d a j e v e o m a k o m p l i k o v a n o . I
m e t o d o m u p o r e i v a a b i i h o v o i z r a q u n a v
a e b i l o v e o m a t e x k o . P r e m a t o m e ,
n a j p r a k t i q n i j e j e p r i m e n i t i f o r m u l u
( 6 ) .
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R a z v i j a e f u n k c i j e u s t e p e n i r e d 4 3
2 . J e d a n p o s t u p a k s u m i r a a s t e p e n i h r e
d o v a
K o e f i c i j e n t i C
i
( i = 0 1 . . . ) u r a z v o j u
1
1
P
i = 0
B
i
x
i
=
1
P
i = 0
C
i
x
i
B
0
6= 0
j e d n o z n a q n o s e o d r e u j u ( n a o s n o v u ( 6 )
) p o m o u f o r m u l a
C
0
=
1
B
0
C
i
= ;
1
B
0
i
P
k = 1
B
k
C
i ; k
i = 1 2 . . . :
O b r n u t o , a k o j e d a t r e d
1
P
i = 0
C
i
x
i
, C
0
6= 0 , k o e f i c i j e n t i B
i
( i = 0 1 . . . ) r e d a
1
P
i = 0
B
i
x
i
j e d n o z n a q n o s e o d r e u j u p o m o u f o r m u l
a
( 7 ) B
0
=
1
C
0
B
i
= ;
1
C
0
i
P
k = 1
C
k
B
i ; k
i = 1 2 . . . :
D e f i n i c i j a . N e k a j e s
n
( x ) =
n
P
i = 0
C
i
x
i
, C
0
6= 0 , i S
n
( x ) =
n
P
i = 0
B
i
x
i
, B
0
6= 0 ,
g d e j e
B
0
=
1
C
0
B
i
= ;
1
C
0
n
P
k = 1
C
k
B
i ; k
i = 1 n :
A k o p o s t o j i l i m S
n
( x ) i a k o j e l i m S
n
( x ) 6= 0 , k a e m o d a j e r e d
1
P
i = 0
C
i
x
i
S
a
- z b i r i v
i d a e g o v S
a
- z b i r i z n o s i
S
a
( x ) = l i m s
n
( x ) =
1
l i m S
n
( x )
:
A k o j e l i m S
n
( x ) = 0 , r e d
1
P
i = 0
C
i
x
i
n i j e S
a
- z b i r i v .
P r i m e r 4 . R e d 1 ; 2 x + 3 x
2
; 4 x
3
+ + ( ; 1 )
i
( i + 1 ) x
i
+ j e S
a
- z b i r i v ,
i e g o v S
a
- z b i r i z n o s i S
a
( x ) =
1
( x + 1 )
2
. O v d e j e
1
P
i = 0
C
i
x
i
= 1 ; 2 x + 3 x
2
; 4 x
3
+ + ( ; 1 )
i
( i + 1 ) x
i
+ :
N a o s n o v u ( 7 ) j e : B
0
= 1 , B
1
= 2 , B
2
= 1 , B
i
= 0 z a i > 3 , j e r j e ( n a o s n o v u ( 7 ) )
B
i
= ; ( ; 1 )
i
( i + 1 ) 1 + ( ; 1 )
i ; 1
i 2 + ( ; 1 )
i ; 2
( i ; 1 ) 1 ] = 0
z a s v e i > 3 , p a j e S
n
( x ) = 1 + 2 x + x
2
= ( 1 + x )
2
, i d a e , n a o s n o v u d e f i n i c i j e
S
a
- z b i r i v o s t i ,
( 8 ) 1 ; 2 x + 3 x
2
; 4 x
3
+ + ( ; 1 )
i
( i + 1 ) x
i
+ =
1
( x + 1 )
2
x 6= ; 1 :
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4 4 B . S a v i
N a p o m e n a 3 . P r i i z r a q u n a v a u k o e f i c i j
e n a t a B
i
( i = 0 1 . . . ) n a o s n o -
v u ( 7 ) , s t e p e n i p r o m e n i v e s e e k s p l i c i
t n o i n e p o j a v u j u u o v o j f o r m u l i , p a
z a k o n k r e t n e v r e d n o s t i p r o m e n i v e s t e
p e n i r e d p r e l a z i u b r o j n i r e d n a k o j i s e
m o e p r i m e n i t i p o s t u p a k S
a
- z b i r i v o s t i .
Z a x = 1 j e d n a k o s t ( 8 ) p o s t a j e
1 ; 2 + 3 ; 4 + + ( ; 1 )
i
( i + 1 ) + =
1
4
:
P r i m e r 5 . A k o r e d 1 ; x + x
3
; x
4
+ x
6
; x
7
+ u p o t p u n i m o q l a n o v i m a
k o j i n e d o s t a j u , d o b i e s e r e d
1 ; x + 0 x
2
+ x
3
; x
4
+ 0 x
5
+ x
6
; x
7
+
p a j e , n a o s n o v u ( 7 ) , B
0
= 1 , B
1
= 1 , B
2
= 1 , B
3
= B
4
= = 0 . D a k l e ,
S
a
( x ) =
1
1 + x + x
2
i d a e , z a x = 1 , 1 ; 1 + 0 + 1 ; 1 + 0 + 1 ; 1 + =
1
3
.
P r i m e r 6 . G e o m e t r i j s k i r e d 1 + q x + q
2
x
2
+ + q
i ; 1
x
i ; 1
+ j e S
a
- z b i r i v ,
S
a
( x ) = 1 + q x + q
2
x
2
+ + q
i ; 1
x
i ; 1
+ =
1
1 ; q x
q x 6= 1 :
N a o s n o v u ( 7 ) j e B
0
= 1 , B
1
= ; q , B
i
= 0 z a i > 2 , p a j e S
a
( x ) =
1
1 ; q x
, q x 6= 1 .
P r i m e r 7 . N a i z b i r r e d a
1
2
2
;
3
2
3
x ;
3
2
4
x
2
; ;
3
2
i + 2
x
i
; :
R e x e e . A k o d a t i r e d n a p i x e m o u o b l i k
u
1
4
;
3
2
3
x
1 +
x
2
+
x
2
2
2
+
( r e d
u z a g r a d i j e g e o m e t r i j s k i r e d ) , t o j
e
S
a
( x ) =
1
4
;
3
2
3
x
1
1 ;
x
2
=
1
2
; x
2 ; x
x 6= 2 :
P r i m e r 8 . H a r m o n i j s k i r e d 1 +
1
2
+
1
3
+ +
1
n
+ n i j e S
a
- z b i r i v .
D a t i r e d n a s t a j e i z r e d a
S
a
( x ) = x +
x
2
2
+
x
3
3
+
x
4
4
+ +
x
i + 1
i + 1
+
z a x = 1 . K a k o j e
S
0
a
( x ) = 1 + x + x
2
+ + x
n
+ =
1
1 ; x
x 6= 1
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R a z v i j a e f u n k c i j e u s t e p e n i r e d 4 5
a o d a v d e S
a
( x ) = ; l n ( 1 ; x ) , x
