. . ( , . ).
. .
, ( ) n = 1,2,..., P(1),P(2),P(3),P(k), , :
1) 1 P(1) ( );
2) k (k ) P(k) k + 1 P(k+1)( );
, . . k (k-).
:
. - . ( ). , n0. n0≠1, .. (1) - . n0 , n0≠1, n0 1 . : n0 1 < n0, n0 = min E, n0 1 , .. P (n01)- , P (n0) ( k P(k) P(k+1)) n0 . .
. , , . (n) , n- . (1), .. (1) , . , () ( + 1). , {n) n.
7.5.
, (1), (2),P(k), , j-. :
1)P(j) ( )
2) k≥j, P(k) , P(k+1)
P(n) n ≥j
=1 ( =0 1).
7.5.
, , n ≥ 4 n! > 2n.
|
|
n, n = 1, n=1 . n = 4, .. n = 2 n = 3.
, , n=4
(n) "n! > 2n".
n = 4. n = 4 4! = 24
24 = 16, 4! > 24.
:
k!> 2k. , (k+1)> 2k+1 , , (k+ 1), 2. , , (k+1)>2, k!> 2k (k+1) > 2 , (k+1)!> 2k+1. k ≥ 4, k > 2. , (k+1)k! > 2 2k (k+1)!> 2k+1. , n! > 2n n > 4.
7.6(1)
, n,
n > .
, , n n≤, . . N . :
β 1 < β, ( n, n > β 1, . . n + 1 > β, n + 1 : /, .