5 200 0 3 2 + 3 2 5 2 3 3 6 2 6 3 + 2 5 6 6 2 + 3 + 2 + 3 2 5 a a b b b a b c d a + c b + d a b, c d 2 3 + 2 + 3 2 5, 2 3 3 5 2 + 3 3 + 5 5 8
2 3 < 2 3 2 5 < 2, 3 5 < 2 3 3 5 5 8 < 2 3. x y y x x < y ( y > x) z x < y x + z < y + z z x < y, z > 0 xz < yz z x < y, z < 0 xz > yz x y x y x y ( y x) a b, c d a b < c d a b < a b c d a + c b + d < c d. a b < c d c d a b a b < a + c b + d a + c b + d a b a b < c d c d a b bc bd ad bd a + c b + d a b b(a + c) + d) a(b b(b + d) b(b + d) bc ad bd b(a + c) a(b + d) b(b + d) 2 > 0. bc ad > 0. ba + bc ab ad b(b + d) bc ad b(b + d) > 0.
b b+d a d c a+c 2 : x x 2 2 x 3 2 2 3, x 4 2 2 3 3 5, x 5 2 3 3 5 5 8, x 6 3 5 5 8 8 3,... x, x 2, x 3, x 4, x 5,... ( n x n ) x n+ x n x n x n a n b n a n+ a n + a n, b n+ b n + b n a n+ b n a 00 a 99 + a 98 a n+ a n a n a n + a n a n+ b n+ b n b n b n + b n a n + a n, b n+ b n + b n a n b n, a n b n 2 a 2 b, a 3 b 2 2 a n + a n b n + b n 2 b n x n b n a n b n b n b n + b n a n+ a n + a n, b n+ b n + b n a n+ b n a n,, 2, 3, 5, 8, 3, 2, 34, 55, 89, 44, 233, 377, 60,... 3
x n a n b n a n a n+, 0.5, 0.6666..., 0.6, 0.625, 0.6538..., 0.6904..., 0.67647..., 0.688..., 0.67977..., 0.68055..., 0.6802575..., 0.680373..., 0.68032786...,... 0.680339888... g + 5 2.68033988... 2: g g 2 g + g g 2 g + g + 5 5 2 x n g a n+ a n + a n x n a n a n+ a n+ a n + a n a n + x n. x n x n x x + x, ( ) 2 x x + g x g2 g 0 g + 5 2 4
3 a b < c d c d a b bc ad bd bc ad bc ad bc ad a b c d > 0 3 5 2 3 2 5 3 5 a b c d a b c d a + c b + d a b a b c d, a b c d c d a b c d 2 p q q b + d a b c d a b c d 2 a b < c d bc ad. a b a + c b + d b(a+c) a(b+d) bc dc a b < p q < c d p q a b < p q bp aq > 0 bp aq, p q < c d cq dp > 0 cq aq. bp aq d cq aq b q (bc ad)q (bp aq)d + (cq aq)b d + b. p q a + c b + d 4 x x p x x q 5
x p q p q x p q π 3.459265358979... 34 00 3.4 22 7 3.4285743... 34592 355 3.4592 3.4592920... 000000 3 22 355 π 7 3 x 2 x 0, x (x x 0 +. ) x 2 x 0 x x 0 x, x 0 x 2, x 2 x x 0 x 2 x 2 x x x 0 x 2 x x 0 < x < x 2 x 3 x 0 x 2 x 2 x x x 3 x 2 x x n x k (k < n) x x n+ x n+ x n x k x n x N N x x x n g x g x n x n x n x n x g x n+ x n x n 0 6
0 2 0 2, 2 2 5 x x [x] [x] [x] x < [x] + [2.3456] 2, [ 7.89] 8. x a m y m (m 0,, 2,... ) a 0 [x] y 0 x a 0, m [ ] a m+ y m y m+ y m a m a m, 0 < y m < (m ). y m a m+ + y m+ y n x a 0 + y 0 a 0 + a 0 + a + a 2 + a + y a 0 +... + a m + y m a m+ + y m+ a + a 2 + y 2 x a 0 + a + a 2 + a 3 +... 7
y n 0 p m p m a 0 + a + a 2 +... + p m x n x n x n p m x N N a x b bx a b a p m b a m 3: x n p m g g 2 g + g + g y m g g, g, a m y m g + + +... 8
g g g + g 2 2 h + 2 a m y m + 2 c + 2d a + 2b a 2b (a + 2b)(a 2b) a 2 2b 2 x h + 2 a 0 2, y 0 ( 2 + ) 2, y 0 2 ( 2 )( 2 + ) 2 + h. ym 2, + 2 h, a m 2. y m h 2 + 2 + 2 +... π 3.4592653589793238... 22 355 7 3 a 0 3, y 0 π 3 0.4592..., 7.0625..., y 0 a 7, y 0.0625..., 5.99659..., a 2 5, y 2 0.99659...,.0034..., y y 2 a 3, y 3 0.0034..., 293.09689..., a 4 293,... y 3 3+ 7 22 7, 3+ 7 + 5 333 06, 3+ 7 + 5 + 355 3, 3+ 7 + 5 + + 293 04348 3325 p m x n 9
x 3 + 3 + 3 +, y + 2 + +. 3 +... 2 +... y 2 6 p q 2q 2 a b c d 4: c d a b bc ad, bd 0
2b 2 2d 2 ( bc ad 2 + b 2 d 2 2b ) 2 2 2d 2 ( 2b + ) 2 ( 2 2d 2 2b ) 2 + 2 2d 2 b 2 d 2 bc ad bc ad >