, , . .
1: . n un≤vn, , .
1: n- S(u)n S(v)n. un≤vn , S(u)n ≤ S(v)n. S2. = S2. , S(v)n< S2 , , S(u)n ≤ S(v)n, S(u)n ≤ S2. , S(u)1, S(u)2, S(u)n, (un>0) S2. {S(u)n} =S1, .. . . , =∞. , S(u)n ≤ S(v)n, =∞, .. .
2 ( ): . , 0, =A (0<A<∞), .
2: n, , , , ε>0 |un/vn-A|<ε, (-ε)vn<un<(A+ε)vn. , (-ε)vn<un<(A+ε)vn 1 , . , 1 , . , (-ε)vn<un<(A+ε)vn, 1, 1, , . , (), () .
.
: =u1+u2++un+
=ℓ. ℓ<1 ℓ>1.
=ℓ, ε>0 N , n>N -
|un+1/un-ℓ|<ε ℓ-ε<un+1/un<ℓ+ε.
ℓ<1. ε , ℓ+ε<1.
ℓ+ε=q, q<1. |un+1/un-ℓ|<ε ℓ-ε<un+1/un<ℓ+ε un+1/un<
q, un+1/<qun,n>N. 3
, un+1/<
qun
n = 1,2,3,... n
, , .. u2+u3+u4++un+ qu1+q2u1+q3u1++qn+1u1+, 0<q<1. , , u2+u3+u4++un+, =u1+u2++un+
|
|
ℓ>1. =ℓ>1. , , N, un+1/un>1, un+1>un, .. n. ≠0. =u1+u2++un+ .
: 1. ℓ=1, =u1+u2++un+ , . 2. , n! an.