:: On the Partial Product of Series and Related Basic Inequalities :: by Fuguo Ge and Xiquan Liang :: :: Received July 6, 2005 :: Copyright (c) 2005-2012 Association of Mizar Users begin registration let x be real number ; cluster|.x.| -> non negative ; coherence not abs x is negative ; end; Lm1: for x being real number holds x ^2 = x |^ 2 proofend; Lm2: 1 |^ 3 = 1 by NEWTON:10; Lm3: 2 |^ 3 = 8 proofend; Lm4: 3 |^ 3 = 27 proofend; Lm5: for x being real number holds (- x) |^ 3 = - (x |^ 3) proofend; Lm6: for x, y being real number holds (x + y) |^ 3 = (((x |^ 3) + ((3 * (x ^2)) * y)) + ((3 * x) * (y ^2))) + (y |^ 3) proofend; Lm7: for x, y being real number holds (x |^ 3) + (y |^ 3) = (x + y) * (((x ^2) - (x * y)) + (y ^2)) proofend; Lm8: for x, y, z being real number st x ^2 <= y * z holds abs x <= sqrt (y * z) proofend; theorem :: SERIES_3:1 for y, x, m being real number st y > x & x >= 0 & m >= 0 holds x / y <= (x + m) / (y + m) proofend; theorem Th2: :: SERIES_3:2 for a, b being real positive number holds (a + b) / 2 >= sqrt (a * b) proofend; theorem :: SERIES_3:3 for b, a being real positive number holds (b / a) + (a / b) >= 2 proofend; theorem Th4: :: SERIES_3:4 for x, y being real number holds ((x + y) / 2) ^2 >= x * y proofend; theorem :: SERIES_3:5 for x, y being real number holds ((x ^2) + (y ^2)) / 2 >= ((x + y) / 2) ^2 proofend; theorem Th6: :: SERIES_3:6 for x, y being real number holds (x ^2) + (y ^2) >= (2 * x) * y proofend; theorem Th7: :: SERIES_3:7 for x, y being real number holds ((x ^2) + (y ^2)) / 2 >= x * y proofend; theorem Th8: :: SERIES_3:8 for x, y being real number holds (x ^2) + (y ^2) >= (2 * (abs x)) * (abs y) proofend; theorem Th9: :: SERIES_3:9 for x, y being real number holds (x + y) ^2 >= (4 * x) * y proofend; theorem Th10: :: SERIES_3:10 for x, y, z being real number holds ((x ^2) + (y ^2)) + (z ^2) >= ((x * y) + (y * z)) + (x * z) proofend; theorem :: SERIES_3:11 for x, y, z being real number holds ((x + y) + z) ^2 >= 3 * (((x * y) + (y * z)) + (x * z)) proofend; theorem Th12: :: SERIES_3:12 for a, b, c being real positive number holds ((a |^ 3) + (b |^ 3)) + (c |^ 3) >= ((3 * a) * b) * c proofend; theorem Th13: :: SERIES_3:13 for a, b, c being real positive number holds (((a |^ 3) + (b |^ 3)) + (c |^ 3)) / 3 >= (a * b) * c proofend; theorem :: SERIES_3:14 for a, b, c being real positive number holds (((a / b) |^ 3) + ((b / c) |^ 3)) + ((c / a) |^ 3) >= ((b / a) + (c / b)) + (a / c) proofend; theorem Th15: :: SERIES_3:15 for a, b, c being real positive number holds (a + b) + c >= 3 * (3 -root ((a * b) * c)) proofend; theorem :: SERIES_3:16 for a, b, c being real positive number holds ((a + b) + c) / 3 >= 3 -root ((a * b) * c) proofend; theorem :: SERIES_3:17 for x, y, z being real number st (x + y) + z = 1 holds ((x * y) + (y * z)) + (x * z) <= 1 / 3 proofend; theorem Th18: :: SERIES_3:18 for x, y being real number st x + y = 1 holds x * y <= 1 / 4 proofend; theorem :: SERIES_3:19 for x, y being real number st x + y = 1 holds (x ^2) + (y ^2) >= 1 / 2 proofend; theorem :: SERIES_3:20 for a, b being real positive number st a + b = 1 holds (1 + (1 / a)) * (1 + (1 / b)) >= 9 proofend; theorem :: SERIES_3:21 for x, y being real number st x + y = 1 holds (x |^ 3) + (y |^ 3) >= 1 / 4 proofend; theorem :: SERIES_3:22 for a, b being real positive number st a + b = 1 holds (a |^ 3) + (b |^ 3) < 1 proofend; theorem :: SERIES_3:23 for a, b being real positive number st a + b = 1 holds (a + (1 / a)) * (b + (1 / b)) >= 25 / 4 proofend; theorem :: SERIES_3:24 for x, a being real number st abs x <= a holds x ^2 <= a ^2 proofend; theorem :: SERIES_3:25 for a being real positive number for x being real number st abs x >= a holds x ^2 >= a ^2 proofend; theorem :: SERIES_3:26 for x, y being real number holds abs ((abs x) - (abs y)) <= (abs x) + (abs y) proofend; theorem :: SERIES_3:27 for a, b, c being real positive number st (a * b) * c = 1 holds ((1 / a) + (1 / b)) + (1 / c) >= ((sqrt a) + (sqrt b)) + (sqrt c) proofend; theorem :: SERIES_3:28 for x, y, z being real number st x > 0 & y > 0 & z < 0 & (x + y) + z = 0 holds (((x |^ 2) + (y |^ 2)) + (z |^ 2)) |^ 3 >= 6 * ((((x |^ 3) + (y |^ 3)) + (z |^ 3)) ^2) proofend; theorem :: SERIES_3:29 for a, b, c being real positive number st a >= 1 holds (a to_power b) + (a to_power c) >= 2 * (a to_power (sqrt (b * c))) proofend; theorem :: SERIES_3:30 for a, b, c being real positive number st a >= b & b >= c holds ((a to_power a) * (b to_power b)) * (c to_power c) >= ((a * b) * c) to_power (((a + b) + c) / 3) proofend; theorem Th31: :: SERIES_3:31 for n being Element of NAT for a, b being real non negative number holds (a + b) |^ (n + 2) >= (a |^ (n + 2)) + (((n + 2) * (a |^ (n + 1))) * b) proofend; theorem :: SERIES_3:32 for a, b being real positive number for n being Element of NAT holds ((a |^ n) + (b |^ n)) / 2 >= ((a + b) / 2) |^ n proofend; theorem Th33: :: SERIES_3:33 for s being Real_Sequence st ( for n being Element of NAT holds s . n > 0 ) holds for n being Element of NAT holds (Partial_Sums s) . n > 0 proofend; theorem Th34: :: SERIES_3:34 for s being Real_Sequence st ( for n being Element of NAT holds s . n >= 0 ) holds for n being Element of NAT holds (Partial_Sums s) . n >= 0 proofend; theorem Th35: :: SERIES_3:35 for n being Element of NAT for s being Real_Sequence st ( for n being Element of NAT holds s . n < 0 ) holds (Partial_Sums s) . n < 0 proofend; theorem Th36: :: SERIES_3:36 for s, s1 being Real_Sequence st s = s1 (#) s1 holds for n being Element of NAT holds (Partial_Sums s) . n >= 0 proofend; theorem :: SERIES_3:37 for n being Element of NAT for s being Real_Sequence st ( for n being Element of NAT holds ( s . n > 0 & s . n > s . (n - 1) ) ) holds (n + 1) * (s . (n + 1)) > (Partial_Sums s) . n proofend; theorem Th38: :: SERIES_3:38 for n being Element of NAT for s being Real_Sequence st ( for n being Element of NAT holds ( s . n > 0 & s . n >= s . (n - 1) ) ) holds (n + 1) * (s . (n + 1)) >= (Partial_Sums s) . n proofend; theorem :: SERIES_3:39 for s, s1, s2 being Real_Sequence st s = s1 (#) s2 & ( for n being Element of NAT holds ( s1 . n >= 0 & s2 . n >= 0 ) ) holds for n being Element of NAT holds (Partial_Sums s) . n <= ((Partial_Sums s1) . n) * ((Partial_Sums s2) . n) proofend; theorem :: SERIES_3:40 for n being Element of NAT for s, s1, s2 being Real_Sequence st s = s1 (#) s2 & ( for n being Element of NAT holds ( s1 . n < 0 & s2 . n < 0 ) ) holds (Partial_Sums s) . n <= ((Partial_Sums s1) . n) * ((Partial_Sums s2) . n) proofend; theorem Th41: :: SERIES_3:41 for s being Real_Sequence for n being Element of NAT holds abs ((Partial_Sums s) . n) <= (Partial_Sums (abs s)) . n proofend; theorem :: SERIES_3:42 for n being Element of NAT for s being Real_Sequence holds (Partial_Sums s) . n <= (Partial_Sums (abs s)) . n proofend; definition let s be Real_Sequence; func Partial_Product s -> Real_Sequence means :Def1: :: SERIES_3:def 1 ( it . 0 = s . 0 & ( for n being Element of NAT holds it . (n + 1) = (it . n) * (s . (n + 1)) ) ); existence ex b1 being Real_Sequence st ( b1 . 0 = s . 0 & ( for n being Element of NAT holds b1 . (n + 1) = (b1 . n) * (s . (n + 1)) ) ) proofend; uniqueness for b1, b2 being Real_Sequence st b1 . 0 = s . 0 & ( for n being Element of NAT holds b1 . (n + 1) = (b1 . n) * (s . (n + 1)) ) & b2 . 0 = s . 0 & ( for n being Element of NAT holds b2 . (n + 1) = (b2 . n) * (s . (n + 1)) ) holds b1 = b2 proofend; end; :: deftheorem Def1 defines Partial_Product SERIES_3:def_1_:_ for s, b2 being Real_Sequence holds ( b2 = Partial_Product s iff ( b2 . 0 = s . 0 & ( for n being Element of NAT holds b2 . (n + 1) = (b2 . n) * (s . (n + 1)) ) ) ); theorem Th43: :: SERIES_3:43 for n being Element of NAT for s being Real_Sequence st ( for n being Element of NAT holds s . n > 0 ) holds (Partial_Product s) . n > 0 proofend; theorem Th44: :: SERIES_3:44 for n being Element of NAT for s being Real_Sequence st ( for n being Element of NAT holds s . n >= 0 ) holds (Partial_Product s) . n >= 0 proofend; theorem :: SERIES_3:45 for s being Real_Sequence st ( for n being Element of NAT holds ( s . n > 0 & s . n < 1 ) ) holds for n being Element of NAT holds ( (Partial_Product s) . n > 0 & (Partial_Product s) . n < 1 ) proofend; theorem :: SERIES_3:46 for s being Real_Sequence st ( for n being Element of NAT holds s . n >= 1 ) holds for n being Element of NAT holds (Partial_Product s) . n >= 1 proofend; theorem :: SERIES_3:47 for s1, s2 being Real_Sequence st ( for n being Element of NAT holds ( s1 . n >= 0 & s2 . n >= 0 ) ) holds for n being Element of NAT holds ((Partial_Product s1) . n) + ((Partial_Product s2) . n) <= (Partial_Product (s1 + s2)) . n proofend; theorem :: SERIES_3:48 for n being Element of NAT for s being Real_Sequence st ( for n being Element of NAT holds s . n = ((2 * n) + 1) / ((2 * n) + 2) ) holds (Partial_Product s) . n <= 1 / (sqrt ((3 * n) + 4)) proofend; Lm9: for s being Real_Sequence st ( for n being Element of NAT holds ( s . n > - 1 & s . n < 0 ) ) holds for n being Element of NAT holds ((Partial_Sums s) . n) * (s . (n + 1)) >= 0 proofend; theorem :: SERIES_3:49 for s1, s being Real_Sequence st ( for n being Element of NAT holds ( s1 . n = 1 + (s . n) & s . n > - 1 & s . n < 0 ) ) holds for n being Element of NAT holds 1 + ((Partial_Sums s) . n) <= (Partial_Product s1) . n proofend; Lm10: for s being Real_Sequence st ( for n being Element of NAT holds s . n >= 0 ) holds for n being Element of NAT holds ((Partial_Sums s) . n) * (s . (n + 1)) >= 0 proofend; theorem :: SERIES_3:50 for s1, s being Real_Sequence st ( for n being Element of NAT holds ( s1 . n = 1 + (s . n) & s . n >= 0 ) ) holds for n being Element of NAT holds 1 + ((Partial_Sums s) . n) <= (Partial_Product s1) . n proofend; theorem :: SERIES_3:51 for s3, s1, s2, s4, s5 being Real_Sequence st s3 = s1 (#) s2 & s4 = s1 (#) s1 & s5 = s2 (#) s2 holds for n being Element of NAT holds ((Partial_Sums s3) . n) ^2 <= ((Partial_Sums s4) . n) * ((Partial_Sums s5) . n) proofend; Lm11: for n being Element of NAT for s, s1, s2 being Real_Sequence st ( for n being Element of NAT holds s . n = ((s1 . n) + (s2 . n)) ^2 ) holds (Partial_Sums s) . n >= 0 proofend; theorem :: SERIES_3:52 for s4, s1, s5, s2, s3 being Real_Sequence st s4 = s1 (#) s1 & s5 = s2 (#) s2 & ( for n being Element of NAT holds ( s1 . n >= 0 & s2 . n >= 0 & s3 . n = ((s1 . n) + (s2 . n)) ^2 ) ) holds for n being Element of NAT holds sqrt ((Partial_Sums s3) . n) <= (sqrt ((Partial_Sums s4) . n)) + (sqrt ((Partial_Sums s5) . n)) proofend; Lm12: for n being Element of NAT for s being Real_Sequence st ( for n being Element of NAT holds s . n > 0 ) holds (n + 1) -root ((Partial_Product s) . n) > 0 proofend; Lm13: for n being Element of NAT for s being Real_Sequence st ( for n being Element of NAT holds ( s . n > 0 & s . n >= s . (n - 1) ) ) holds ((s . (n + 1)) - (((Partial_Sums s) . n) / (n + 1))) / (n + 2) >= 0 proofend; theorem :: SERIES_3:53 for n being Element of NAT for s being Real_Sequence st ( for n being Element of NAT holds ( s . n > 0 & s . n >= s . (n - 1) ) ) holds (Partial_Sums s) . n >= (n + 1) * ((n + 1) -root ((Partial_Product s) . n)) proofend;