a+b*i, , i (: , -1).
(Sn) (Sbni) , . ( .)
7. : , . (!! = ).
S(x),ÎW , S(x) =lim n→∞ S(x), S(x)= f 1(x)+ f 2(x)++ fn (x)
S(x), ÎL (LÍΩ) f 1(x)+ f 2(x)++ fn (x)+=n=1∑ ∞ fn (x) ( ), , L S(x).
L S(x), e>0 N , n³N c ÎL ½S(x) -S n (x)½<e
L, ,
L .
.
f 1(x)+ f 2(x)++ fn (x)+ L ½ fn (x)½≤n (n=1,2), n 1+2++ n+ L .
:
fn (x) [a,b], f 1(x)+ f 2(x)++ fn (x)+,
1. f (x) [a,b]
2. a∫ b f (x)dx=. a∫ b f 1(x)dx++. a∫ b fn (x) dx+
fn (x) [a,b]
) f 1(x)+ f 2(x)++ fn (x)+ f (x)
) f 1 ' (x)+ f 2 ' (x)++ fn' (x)+ , f (x) f ' (x)= f 1 ' (x)+ f 2 ' (x)++ fn' (x)+
.
. 0+1+22+++, (*)
0, 1,2, - .
0,1,2,- .
, . , . , (*) , .
.
1) (*) 0≠0, , : ||<|0|.
2) (*) . 1≠0, x: ||>|1|.
-.1). 0+10+202++0+(**) , 0 →0, →∞. , {0}
, .. - , |0|<M =0,1,2
|0|+|10|+|202|++|0|+.(***)
|
|
||<|0|, ||=|0||/0|<|/0|, |/0|<1. (***)
+|/0|+|/0|2++|/0|+- . (***) , (**) .
2), (**) =1, :| |>1 (**) =1, .