series of functions
TRANSCRIPT
-
8/13/2019 Series of Functions
1/236
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8/13/2019 Series of Functions
2/236
http://find/ -
8/13/2019 Series of Functions
3/236
,
, . . . ,
,. . .
,
( ),
( ), . . . , ,( ). . .
{
} .
N, =R
( ) =
+
( ) = ( +
)
( ) = ( +
)
( ) = (
+
)
. . .
( ) = ( +
)
http://find/ -
8/13/2019 Series of Functions
4/236
,
, . . . ,
,. . .
,
( ),
( ), . . . , ,( ). . .
{
} .
N, =R
( ) =
+
( ) = ( +
)
( ) = ( +
)
( ) = (
+
)
. . .
( ) = ( +
)
http://find/ -
8/13/2019 Series of Functions
5/236
,
, . . . ,
,. . .
,
( ),
( ), . . . , ,( ). . .
{
} .
N, =R
( ) =
+
( ) = ( +
)
( ) = ( +
)
( ) = (
+
)
. . .
( ) = ( +
)
http://find/ -
8/13/2019 Series of Functions
6/236
,
, . . . ,
,. . .
,
( ),
( ), . . . , ,( ). . .
{
} .
N, =R
( ) =
+
( ) = ( +
)
( ) = ( +
)
( ) = (
+
)
. . .
( ) = ( +
)
http://find/ -
8/13/2019 Series of Functions
7/236
,
, . . . ,
,. . .
,
( ),
( ), . . . , ,( ). . .
{
} .
N, =R
( ) =
+
( ) = ( +
)
( ) = ( +
)
( ) = (
+
)
. . .
( ) = ( +
)
http://find/http://goback/ -
8/13/2019 Series of Functions
8/236
,
, . . . ,
,. . .
,
( ),
( ), . . . , ,( ). . .
{
} .
N, =R
( ) =
+
(
) = (
+
)
( ) = ( +
)
( ) = (
+
)
. . .
( ) = ( +
)
http://find/ -
8/13/2019 Series of Functions
9/236
,
, . . . ,
,. . .
,
( ),
( ), . . . , ,( ). . .
{
} .
N, =R
( ) =
+
(
) = (
+
)
( ) = ( +
)
( ) = (
+
)
. . .
( ) = ( +
)
http://goforward/http://find/http://goback/ -
8/13/2019 Series of Functions
10/236
,
,
,
,
( ) = ( +
) .
http://find/ -
8/13/2019 Series of Functions
11/236
{
}
{
( )} . {
}
( ) =
( ),
.
{ } > , > |
( ) ( )| > .
http://find/ -
8/13/2019 Series of Functions
12/236
{
}
{
( )} . {
}
( ) =
( ),
.
{ } > , > |
( ) ( )| > .
http://find/ -
8/13/2019 Series of Functions
13/236
-
8/13/2019 Series of Functions
14/236
( ) = , R+, N.
( ) =
( ) = = = .
=
;
( ) =
, [ ,), N.
( ) = [ ,).
http://find/ -
8/13/2019 Series of Functions
15/236
( ) = , R+, N.
( ) =
( ) = = = .
=
;
( ) =
, [ ,), N.
( ) = [ ,).
http://find/ -
8/13/2019 Series of Functions
16/236
( ) = , R+, N.
( ) =
( ) = = = .
=
;
( ) =
, [ ,), N.
( ) = [ ,).
http://goforward/http://find/http://goback/ -
8/13/2019 Series of Functions
17/236
( ) =
+
[ ,+) .
<
( ) =
+
( ) = < ;
=
( ) =
/
/ ;
>
( ) =
+
=
+ / > .
{
}
( ) =
[ , ) / =
( ,+)
http://find/ -
8/13/2019 Series of Functions
18/236
( ) =
+
[ ,+) .
<
( ) =
+
( ) = < ;
=
( ) =
/
/ ;
>
( ) =
+
=
+ / > .
{
}
( ) =
[ , ) / =
( ,+)
http://find/ -
8/13/2019 Series of Functions
19/236
( ) =
+
[ ,+) .
<
( ) =
+
( ) = < ;
=
( ) =
/
/ ;
>
( ) =
+
=
+ / > .
{
}
( ) =
[ , ) / =
( ,+)
http://find/ -
8/13/2019 Series of Functions
20/236
( ) =
+
[ ,+) .
<
( ) =
+
( ) = < ;
=
( ) =
/
/ ;
>
( ) =
+
=
+ / > .
{
}
( ) =
[ , ) / =
( ,+)
http://goforward/http://find/http://goback/ -
8/13/2019 Series of Functions
21/236
( ) =
+
[ ,+) .
<
( ) =
+
( ) = < ;
=
( ) =
/
/ ;
>
( ) =
+
=
+ / > .
{
}
( ) =
[ , ) / =
( ,+)
http://find/ -
8/13/2019 Series of Functions
22/236
( ) =
+
[ ,+) .
<
( ) =
+
( ) = < ;
=
( ) =
/
/ ;
>
( ) =
+
=
+ / > .
{
}
( ) =
[ , ) / =
( ,+)
http://find/ -
8/13/2019 Series of Functions
23/236
=
: , Z, , ( , ) = .
( ) =
,
/
.
/ , ( ) = N
>
,
( ) =
>
.
( ) =
,
.
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8/13/2019 Series of Functions
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=
: , Z, , ( , ) = .
( ) =
,
/
.
/ , ( ) = N
>
,
( ) =
>
.
( ) =
,
.
http://goforward/http://find/http://goback/ -
8/13/2019 Series of Functions
25/236
=
: , Z, , ( , ) = .
( ) =
,
/
.
/ , ( ) = N
>
,
( ) =
>
.
( ) =
,
.
http://find/ -
8/13/2019 Series of Functions
26/236
=
: , Z, , ( , ) = .
( ) =
,
/
.
/ , ( ) = N
>
,
( ) =
>
.
( ) =
,
.
http://goforward/http://find/http://goback/ -
8/13/2019 Series of Functions
27/236
( )
,
( , ) = {| ( ) ( )|: }:= , .
,
( ).
{
}
( )
( )
,
(
, ) =
{|
( ) ( )|: }= .
> , >
, ( , ) = {| ( ) ( )|: }:= ,
.
{
}
,
(
, ) = .
http://goforward/http://find/http://goback/ -
8/13/2019 Series of Functions
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-
8/13/2019 Series of Functions
29/236
( )
,
( , ) = {| ( ) ( )|: }:= , .
,
( ).
{
}
( )
( )
,
(
, ) =
{|
( ) ( )|: }= .
> , >
, ( , ) = {| ( ) ( )|: }:= ,
.
{
}
,
(
, ) = .
http://goforward/http://find/http://goback/ -
8/13/2019 Series of Functions
30/236
( )
,
( , ) = {| ( ) ( )|: }:= , .
,
( ).
{
}
( )
( )
,
(
, ) =
{|
( ) ( )|: }= .
> , >
, ( , ) = {| ( ) ( )|: }:= ,
.
{
}
,
(
, ) = .
http://find/ -
8/13/2019 Series of Functions
31/236
Uniform Convergence does not depend on the choice of the variable
in the domain of convergence.
( ) =
, [ ,), N.
,[ ,)( , ) =
,
[
,).
( ) = , R+, N,
[ ,)
http://goforward/http://find/http://goback/ -
8/13/2019 Series of Functions
32/236
Uniform Convergence does not depend on the choice of the variable
in the domain of convergence.
( ) =
, [ ,), N.
,[ ,)( , ) =
,
[
,).
( ) = , R+, N,
[ ,)
http://find/ -
8/13/2019 Series of Functions
33/236
Uniform Convergence does not depend on the choice of the variable
in the domain of convergence.
( ) =
, [ ,), N.
,[ ,)( , ) =
,
[
,).
( ) = , R+, N,
[ ,)
http://goforward/http://find/http://goback/ -
8/13/2019 Series of Functions
34/236
Uniform Convergence does not depend on the choice of the variable
in the domain of convergence.
( ) =
, [ ,), N.
,[ ,)( , ) =
,
[
,).
( ) = , R+, N,
[ ,)
http://find/ -
8/13/2019 Series of Functions
35/236
Uniform Convergence does not depend on the choice of the variable
in the domain of convergence.
( ) =
, [ ,), N.
,[ ,)( , ) =
,
[
,).
( ) = , R+, N,
[ ,)
http://find/ -
8/13/2019 Series of Functions
36/236
Uniform Convergence does not depend on the choice of the variable
in the domain of convergence.
( ) =
, [ ,), N.
,[ ,)( , ) =
,
[
,).
( ) = , R+, N,
[ ,)
http://goforward/http://find/http://goback/ -
8/13/2019 Series of Functions
37/236
( ) = ( )
R.
> . R
|
( ) |=
( )
=
( ) = = /. > ,
,R( , ) = {| ( ) ( )|: R}=
. R
|
( ) |=
( )
=
( ) = = /. > ,
,R( , ) = {| ( ) ( )|: R}=
. R
|
( ) |=
( )
=
( ) = = /. > ,
,R( , ) = {| ( ) ( )|: R}=
. R
|
( ) |=
( )
=
( ) = = /. > ,
,R( , ) = {| ( ) ( )|: R}=
/
for every
,and hence ,[ , ]( , ) .
( ) =
, N
{
}
[ ,
]
<
< ,
,[
,
]( , ) = {| |: [ , ]}= .
http://goforward/http://find/http://goback/ -
8/13/2019 Series of Functions
45/236
.
,[ , ](
,
)>
/
for every
,and hence ,[ , ]( , ) .
( ) =
, N
{
}
[ ,
]
<
< ,
,[
,
]( , ) = {| |: [ , ]}= .
http://find/ -
8/13/2019 Series of Functions
46/236
.
,[
,
](
,
)>
/
for every
,and hence ,[ , ]( , ) .
( ) =
, N
{
}
[ ,
]
<
< ,
,[
,
]( , ) = {| |: [ , ]}= .
http://goforward/http://find/http://goback/ -
8/13/2019 Series of Functions
47/236
.
,[
,
](
,
)>
/
for every
,and hence ,[ , ]( , ) .
( ) =
, N
{
}
[ ,
]
<
< ,
,[
,
]( , ) = {| |: [ , ]}= .
http://find/ -
8/13/2019 Series of Functions
48/236
.
,[
,
](
,
)>
/
for every
,and hence ,[ , ]( , ) .
( ) =
, N
{
}
[ ,
]
<
< ,
,[
,
]( , ) = {| |: [ , ]}= .
http://find/ -
8/13/2019 Series of Functions
49/236
.
,[
,
](
,
)>
/
for every
,and hence ,[ , ]( , ) .
( ) =
, N
{
}
[ ,
]
<
< ,
,[
,
]( , ) = {| |: [ , ]}= .
http://goforward/http://find/http://goback/ -
8/13/2019 Series of Functions
50/236
.
,[
,
](
,
)>
/
for every
,and hence ,[ , ]( , ) .
( ) =
, N
{
}
[ ,
]
<
< ,
,[
,
]( , ) = {| |: [ , ]}= .
http://find/ -
8/13/2019 Series of Functions
51/236
{
} > >
, ( , + ) = + , N.
{
} .
> , > , ( , ) = , .
>
N
, ( , + ) , ( , ) + , ( , + )
-
8/13/2019 Series of Functions
52/236
{
} > >
, ( , + ) = + , N.
{
} .
> , > , ( , ) = , .
>
N
, ( , + ) , ( , ) + , ( , + )
-
8/13/2019 Series of Functions
53/236
{
} > >
, ( , + ) = + , N.
{
} .
> , > , ( , ) = , .
>
N
, ( , + ) , ( , ) + , ( , + ) >
http://find/http://goback/ -
8/13/2019 Series of Functions
54/236
, ( , + ) N.
, > > | ( ) + ( )| N. {
( )}
( ).
>
,
N
|
( ) ( )| |
( )
+
( )| + |
+
( ) ( )|
, ( , + ) + | + ( ) ( )|
>
http://goforward/http://find/http://goback/ -
8/13/2019 Series of Functions
55/236
, ( , + ) N.
, > > | ( ) + ( )| N. {
( )}
( ).
>
,
N
|
( ) ( )| |
( )
+
( )| + |
+
( ) ( )|
, ( , + ) + | + ( ) ( )|
>
http://goforward/http://find/http://goback/ -
8/13/2019 Series of Functions
56/236
, ( , + ) N.
, > > | ( ) + ( )| N. {
( )}
( ).
>
,
N
|
( ) ( )| |
( )
+
( )| + |
+
( ) ( )|
, ( , + ) + | + ( ) ( )|
>
http://find/ -
8/13/2019 Series of Functions
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, ( , + ) N.
, > > | ( ) + ( )| N. {
( )}
( ).
>
,
N
|
( ) ( )| |
( )
+
( )| + |
+
( ) ( )|
, ( , + ) + | + ( ) ( )|
>
http://goforward/http://find/http://goback/ -
8/13/2019 Series of Functions
58/236
, ( , + ) N.
, > > | ( ) + ( )| N. {
( )}
( ).
>
,
N
|
( ) ( )| |
( )
+
( )| + |
+
( ) ( )|
, ( , + ) + | + ( ) ( )|
>
http://goforward/http://find/http://goback/ -
8/13/2019 Series of Functions
59/236
, ( , + ) N.
, > > | ( ) + ( )| N. {
( )}
( ).
>
,
N
|
( ) ( )| |
( )
+
( )| + |
+
( ) ( )|
, ( , + ) + | + ( ) ( )|
-
8/13/2019 Series of Functions
60/236
R
| ( )| < .
R.
{
}
(
) =
(
(
))
{
} R
, R
|
(
)
(
)|
|
| .
(R).
|
( )|=|
(
) +
(
( )) ( )|
|
(
)| + |
(
(
))
(
)|| ( )| + |
( ) || ( )| + |
( )| .
R,
.
http://find/ -
8/13/2019 Series of Functions
61/236
R
| ( )| < .
R.
{
}
(
) =
(
(
))
{
} R
, R
|
(
)
(
)|
|
| .
(R).
|
( )|=|
(
) +
(
( )) ( )|
|
(
)| + |
(
(
))
(
)|| ( )| + |
( ) || ( )| + |
( )| .
R,
.
http://find/ -
8/13/2019 Series of Functions
62/236
R
| ( )| < .
R.
{
}
(
) =
(
(
))
{
} R
, R
|
(
)
(
)|
|
| .
(R).
|
( )|=|
(
) +
(
( )) ( )|
|
(
)| + |
(
(
))
(
)|| ( )| + |
( ) || ( )| + |
( )| .
R,
.
http://goforward/http://find/http://goback/ -
8/13/2019 Series of Functions
63/236
R
| ( )| < .
R.
{
}
(
) =
(
(
))
{
} R
, R
|
(
)
(
)|
|
| .
(R).
|
( )|=|
(
) +
(
( )) ( )|
|
(
)| + |
(
(
))
(
)|| ( )| + |
( ) || ( )| + |
( )| .
R,
.
{
}
(R).
http://find/ -
8/13/2019 Series of Functions
64/236
{ } ( ) > ,
|
+
(
)
(
)|=|
(
(
))
(
(
))|
|
(
)
(
)|.
,R( + , ) ,R( , )
,R( , )
,R( , ).
,R( + , ) ,R( + , + ) + + ,R( + , )
( + + + ) ,R( , )
,R(
,
).
|
( )
( )|<
| |
-
8/13/2019 Series of Functions
75/236
= [
,
].
[ , ]
( ) =
( ) .
|
( )
( ) |=|
( ( )
( )) |
| ( )
( )
,
(
, ) = ( ) ,
(
, )
http://find/ -
8/13/2019 Series of Functions
76/236
= [
,
].
[ , ]
( ) =
( ) .
|
( )
( ) |=|
( ( )
( )) |
| ( )
( )
,
(
, ) = ( ) ,
(
, )
http://find/ -
8/13/2019 Series of Functions
77/236
= [
,
].
[ , ]
( ) =
( ) .
|
( )
( ) |=|
( ( )
( )) |
| ( )
( )
,
(
, ) = ( ) ,
(
, )
( ) ( )
http://find/ -
8/13/2019 Series of Functions
78/236
( ) =
+
+
+ + ( , ).
/
(
)
.
(
,
),
( ) =
+
( ) =
.
, , |
( ) ( )|=
+
= +
.
, ,
,
,
.
( ) =
= .
( ) + + + + ( )
http://find/ -
8/13/2019 Series of Functions
79/236
( ) =
+
+
+ + ( , ).
/
(
)
.
(
,
),
( ) =
+
( ) =
.
, , |
( ) ( )|=
+
= +
.
, ,
,
,
.
( ) =
= .
( ) + + + + ( )
http://find/ -
8/13/2019 Series of Functions
80/236
( ) =
+
+
+ + ( , ).
/
(
)
.
(
,
),
( ) =
+
( ) =
.
, , |
( ) ( )|=
+
= +
.
, ,
,
,
.
( ) =
= .
( ) + + + + ( )
http://find/ -
8/13/2019 Series of Functions
81/236
( ) =
+
+
+ + ( , ).
/
(
)
.
(
,
),
( ) =
+
( ) =
.
, , |
( ) ( )|=
+
= +
.
, ,
,
,
.
( ) =
= .
( ) = + + + + ( )
http://goforward/http://find/http://goback/ -
8/13/2019 Series of Functions
82/236
( ) =
+
+
+ + ( , ).
/
(
)
.
(
,
),
( ) =
+
( ) =
.
, , |
( ) ( )|=
+
= +
.
, ,
,
,
.
( ) =
= .
http://goforward/http://find/http://goback/ -
8/13/2019 Series of Functions
83/236
= [
,
]
[ , ] ( ) =
( ).
{
}
( ) =
[ , ] < < < .
( ) =
( )
(
,
).
http://find/ -
8/13/2019 Series of Functions
84/236
= [
,
]
[ , ] ( ) =
( ).
{
}
( ) =
[ , ] < < < .
( ) =
( )
(
,
).
http://find/ -
8/13/2019 Series of Functions
85/236
= [
,
]
[ , ] ( ) =
( ).
{
}
( ) =
[ , ] < < < .
( ) =
( )
(
,
).
http://goforward/http://find/http://goback/ -
8/13/2019 Series of Functions
86/236
{ } [ , ].
[
,
]
{
} [ , ];
[ , ] {
(
)}
{
} [ , ]
( ) =
( ),
.
=
( )
=
|
| / < |
| / < ;
| | / = ;
http://find/ -
8/13/2019 Series of Functions
192/236
|
| = ;
|
| / = .
R | |<
R | |> .
http://goforward/http://find/http://goback/ -
8/13/2019 Series of Functions
193/236
=
( )
( , + ) , ( , + ], [ , + ), [ , + ]
-
8/13/2019 Series of Functions
194/236
| |< < < + .
=
( )
( , + ) , ( , + ], [ , + ), [ , + ]
http://find/ -
8/13/2019 Series of Functions
195/236
| |< < < + .
=
(
+
)
.
( + ) + /( + )( + ) /
=
+ | + |=| + | .
http://find/ -
8/13/2019 Series of Functions
196/236
( + ) /
+
| + |< , < .
=
,
=
(
)
,
= , =
.
=
(
+
)
[ , ) .
=
(
+
)
.
( + ) + /( + )( + ) /
=
+ | + |=| + | .
http://goforward/http://find/http://goback/ -
8/13/2019 Series of Functions
197/236
( + ) /
+
| + |< , < .
=
,
=
(
)
,
= , =
.
=
(
+
)
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8/13/2019 Series of Functions
198/236
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204/236
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205/236
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206/236
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207/236
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8/13/2019 Series of Functions
208/236
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8/13/2019 Series of Functions
209/236
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8/13/2019 Series of Functions
210/236
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8/13/2019 Series of Functions
211/236
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