Σειρές
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description
Σειρές
Transcript of Σειρές
-
1 pi 3
.
: 0,999=1.
pi ([email protected]) H pi . (). pi . 1. (pi ) ( ) (=1,2,3,).
( ) : 1 21
... , 1,2,3,...
=
= = + + + =
: ( ) ( )( ), , (pi )
1
+
= ( ,
: 1 2 ... , 1,2,3,... = + + + = ).
( ) 1
+
= .
( ) 1
+
= .
, pi 1
+
= ,
: 1 21
...
=
= = + + + , pi ( ) ,
. ( )
1
+
= .
1
+
= : 1 2 ... ... + + + + pi : 1 2 ... + +
(pi pi pipi),: ( )
1 1, N
+ +
= =
= =
2. 1
+
= pi { },R R = + ,
, ( ) pi . , , : lim = ( pi , , : lim + , : lim = ). : 1 2 3 ... + + + pi , , , (, = + , pi = pi ). pi, , pi .
pipi pi,
( pipi) , pipi pi pi , .. pi pi, pi
pi .
( pi ), : lim = . :
-
2 pi 3
1 2 ... + + pi , . pi, , : : 1 2 ... + +
, lim = , , pi :
1
+
=
= 1 2 3 ... + + + = . (1) pi
, pi ,
{ },R R= + . . pi, pi pipi pi (1) 10. , pi, : lim = , pi: 10lim 10 = , pi pi pi : 1 2 310 10 10 ... + + + 10 pi : 1 2 310 10 10 ... 10 + + + = . , ( , A - 2010- 136) . , pi pi pi , pi pi
( pipi pi pi pi pipi). . pi (1) (pi =). , pi, pi, pi :
1
1 12
+
=
= (2) 1
1 1( 1) +
=
=+ (3), :
1 1
1 12 ( 1)
+ +
= =
=+ (4) :
1 12 ( 1) = + , N
, pi. ,
pi (2) (3) ( pi pi pipi), pi pi, pi (4). (2) (3) pi ()
1
12
+
=
1
1( 1)
+
= + .
pi . pi.
: 0,999=1. pi , : 0,999, pi , :
2 39 9 9
...
10 10 10+ + +
: 2 3
1 19 9 9 9 9 110lim lim ... lim . lim 1 1110 10 10 10 10 10110
= + + + + = = =
.
, : 0,999 1. : 0,999=1. pi pi , , 0,999 1 ( ). pi , pi pi pipi, :
2 39 9 9 9lim ... 1
10 10 10 10
+ + + + =
.
---- , :
1 20 20
...
10 10 10y y yy
+
= = + + +
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3 pi 3
pi: 0y Z { } *0,1,2,3,4,5,6,7,8,9 ,y ( ), . pi pi: 1. pi ( pi ). 2. pi , pi :
1 20 20
...
10 10 10y y yy
+
= = + + + ,
pi: 0y Z { } *0,1,2,3,4,5,6,7,8,9 ,y , pi ( ). pipi pi
* 9y = , >, .
pi : 1 20 2 ...10 10y yy + + + , pi
. : 0 1 20, ...y y y+ : 0 1 2 3, ...y y y y ,
0 0y . pi, , pi pi pipi, : 0 1 2, ...y y y = ( 0 0y ), pi 0 1 2, ...y y y 0y pi : 1 2 3, , ,...y y y ( , ). , pi , : 1 2 ... + + + , pi , (pi ) . : :
, =1,2,...,0,
= >
: 1 2 1 21
... lim( ... )
+
=
+ + + = + + + = , :
1 2 1 2... 0 0 ... ... + + + + + + = + + + . (5). : 1 2 ... + + + ( pi pi , =1) . , pi pipi, (5) pi :
( )1 2lim ... + + + = 1 2 ... + + + pi , : 1 2 ... 0 0 ... + + + + + + (): 1 2 ... + + + . pi . pipi ,
(, pi pi ).
1. pipi .: (1962) 2. pi .: , 1,- (1976) 3. .: (2003) 4. .: (2005) 5. pi .: (1993) 6. Anderson K. and Hall D.: ELEMENTARY REAL ANALYSIS (McGraw-Hill Book Comp.) 7. Gaskill H.-Narayanaswami P.: FOUNDATIONS OF ANALYSIS (Harper & Row,New York) 8. Goldberg R.: Methods of Real Analysis (Blaisdell Publishing Company) 9. Kirkwood J.: An Introduction to Analysis (PWS-KENT, Boston) 10. Maddox I.: Introductory Mathematical Analysis (Adam Hilger Ltd, Bristol) 11. Protter M.-Morrey C.: A First Course in Real Analysis (Springer-Verlag, New York-Berlin) 12. Olmsted J.: THE REAL NUMBER SYSTEM (APPLETON-CENTURY-CROFTS-New York)
