, : f (x) Ì Rn , :
ó g1(x)³0,
î .
ì g s(x)³0
ì g s(x)³0
î x³0
g1(),, g s(x) . F(x,l)= f (x)+l1 g1(x)++l s g s(x), l=(l1,,l s) . , ³0, . *³0 f (x) , l*=(l*1,,l*s)³0, , :
gradxF(x*, l*)£0;
(gradxF(x*, l*);*)=0
gradlF(x*, l*)³0
(gradlF(x*, l*);l*)=0
, (x*, l*) F(x, l), .. F(x, l*)£ F(x*, l*)£ F(x*, l)
.
. . . .
- , : 1+2++ +=∑=1∞.
1,,-
. : S = ∑=1i = 1+2++ - n- : =1, S=1,=2, S=1+2,: S = 1+2++, .. , . .
, S, - , S: LimSk=S, k→∞. .
- . .
2 :
1+2++ +=∑=1∞. (1)
1+2++ +=∑=1∞ (2)
.
1). . (1) (2).
2) , (1) , λ.
-.
1) (2) , S, , λS.
-: Sk- (2), sk - λ 1+ λ 2++λ +, , λ Sk = sk. , :
Lim sk=lim λSk= λlimSk= λS(k→∞)
2) (1) (2) , S, S, 1) ( (3)) , S+S.
-: Qk=Sk+Tk, Qk, Sk,Tk (1), (2), (3). k→∞, , - LimQk Q=S+T
3) , , .
|
|
-: , n . n+1 +n+2+ (1). n- n , Sk - . ,Sk - , k>n:
Sk = Cn+Sk
- lim Sk k→∞, - lim Sk . , : S=S+Cn
4) (1) , . , .
- .
. (1) , →∞ 0. lim ak=0
-.
1){Sk=a1+a2++ak
{Sk-1=a1+a2++ak-1, =Sk-Sk-1
2) , lim Sk = S, k→∞
3) k→∞: lim ak= lim Sk- lim Sk-1 = S- lim Sk-1= S-S=0 ((k-1)→∞)
: lim ak≠0 -, .
. - , .
. (, , )
a1 + a2 + + an + = n=1S¥ an = Sn , . .
S1 = a1 > 0, S2 = a1 + a2> 0, {Sn}-
.
, , .
:
u1 + u2 + + un + = n=1S¥ un, un > 0 " n
v1 + v2 + + vn + = n=1S¥ vn, vn > 0 " n
1) "n Î N: un £ vn n=1S¥ vn , n=1S¥ un .
"n Î N: un £ vn n=1S¥ un , n=1S¥ vn .
2) $ lim un/vn = k, ,
n ¥ k = const
.
.
n=1S¥ un , lim un+1/un = L,
n ¥
1) L < 1
2) L > 1
3) L = 1 .
.
. n=1S¥un - , 1) un= f(n); 2) y = f(x) " x ³ 1, , , 1∫+¥f(x)dx, ,
n=1S¥ un = 1∫+¥f(x)dx