(1)
(uk)kÎN , . S1 = u1, S2 = u1 + u2,.... Sn = u1 + u2 +...+ un, (1).
(2)
(1) , S (1)
. (1) ,
.
u1+u2++un , =S. =S, n (n-1) . , - = = un=0, .
. (1) , , ε > 0 N = N(ε) , n > N = 1, 2,
:
26 . . .
, .
, . . .
. , ( ) : S(n + 1) − S(n) = a(n + 1) , .
.
(17)
(18)
(17) (18), .. (n = 1, 2, 3, ). , (18), (17). (17) , (18) . , n, n = N.
27 .
(7)
p<1 (7) , p>1 (7) .
. . , N, n³N
p-E< (10)
p<1. , p+E=q<1. n³N
(11)
(12)
. (13)
(13) , . (12) , (11), . (7) 1.
p>1. , p-E>1. (10) , n³N un+1>un, n. un¹0, , . .
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.
1. , un¹0.
2. =1 . .
3. .
u1+u2++un (7)
p<1 (7) , p>1 (7) .
. , N, n³N | | <E
p-E< <p+E. (14)
p<1. , p+E=q<1. (14) <q un<qn n³N.
(15)
(16)
(16) , . (15) , , un<qn n³N, , , 1 (7).
p>1. ,
p-E >1. (14) >1 un>1, , un¹0 (7) . .
28 . . . . |
, , , . u1-u2+u3-u4++ +(-1)n-1.un+, un , . , , , . , , , , , . (19) : u1>u2>>un>, (19) , . . S2n=(u1-u2)+(u3-u4)++(u2n-1-u2n). u1>u2>>u2n-1>u2n, , , S2n n S2n>0 n. S2n=u1-[(u2-u3)+(u4-u5)++(u2n-2-u2n-1)+u2n]. S2n>0, S2n<u1 n. , S2n , , S2n=S. 0<S≤u1. S2n+1=S2n+u2n+1. n→∞: S2n+1= S2n+ u2n+1=S+0=S. , , S, Sn=S, . . . 1. un>un+1 , N. 2. un>un+1 . , . |
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