:: SERIES_2 semantic presentation
theorem Th1: :: SERIES_2:1
Lemma26:
for a being real number holds a |^ 3 = (a * a) * a
Lemma27:
for a being real number holds a |^ 3 = (a |^ 2) * a
Lemma28:
for a, b being real number holds
( 4 = 2 |^ 2 & (a + b) |^ 2 = ((a |^ 2) + ((2 * a) * b)) + (b |^ 2) )
Lemma29:
for a being real number holds (a + 1) |^ 3 = (((a |^ 3) + (3 * (a |^ 2))) + (3 * a)) + 1
Lemma30:
for n being Element of NAT holds ((n * ((2 * n) + 1)) * (((3 * (n |^ 2)) + (3 * n)) - 1)) + (((n + 1) |^ 3) * 30) = ((n + 2) * ((2 * (n + 1)) + 1)) * (((3 * ((n + 1) |^ 2)) + (3 * (n + 1))) - 1)
Lemma31:
for n being real number holds
( (n + 1) |^ 4 = ((((n |^ 4) + (4 * (n |^ 3))) + (6 * (n |^ 2))) + (4 * n)) + 1 & (n + 1) |^ 5 = (((((n |^ 5) + (5 * (n |^ 4))) + (10 * (n |^ 3))) + (10 * (n |^ 2))) + (5 * n)) + 1 )
theorem Th2: :: SERIES_2:2
Lemma32:
for n being Element of NAT holds
( 1 - (1 / (n + 2)) = (n + 1) / (n + 2) & 1 / ((n + 1) ! ) = (n + 2) / (((n + 1) ! ) * (n + 2)) )
Lemma33:
for n being Element of NAT holds (1 / (n + 1)) - (2 / ((n + 1) * (n + 3))) = 1 / (n + 3)
Lemma34:
for n being Element of NAT holds (1 / (n + 1)) - (3 / ((n + 1) * (n + 4))) = 1 / (n + 4)
Lemma35:
for n being Element of NAT holds ((n |^ 2) + (3 * n)) + 2 = (n + 1) * (n + 2)
Lemma36:
for n being Element of NAT holds (((n * (4 * (n |^ 2))) + (11 * n)) + (12 * (n |^ 2))) + 3 = (n + 1) * ((4 * ((n + 1) |^ 2)) - 1)
Lemma37:
for n being Element of NAT holds ((n + 1) |^ 2) * ((2 * ((n + 1) |^ 2)) - 1) = ((((((n |^ 2) * (n |^ 2)) * 2) + ((12 - 1) * (n |^ 2))) + (8 * (n |^ 3))) + (6 * n)) + 1
Lemma38:
for n being Element of NAT holds (n + 2) * ((((n + 1) |^ 2) + (n + 1)) - 1) = (((n |^ 3) + (5 * (n |^ 2))) + (7 * n)) + 2
Lemma39:
for a, b being real number
for s being Real_Sequence st ( for n being Element of NAT holds s . n = (a * n) + b ) holds
for n being Element of NAT holds (Partial_Sums s) . n = ((((a * (n + 1)) * n) / 2) + (n * b)) + b
theorem Th3: :: SERIES_2:3
theorem Th4: :: SERIES_2:4
theorem Th5: :: SERIES_2:5
theorem Th6: :: SERIES_2:6
theorem Th7: :: SERIES_2:7
theorem Th8: :: SERIES_2:8
theorem Th9: :: SERIES_2:9
theorem Th10: :: SERIES_2:10
theorem Th11: :: SERIES_2:11
theorem Th12: :: SERIES_2:12
theorem Th13: :: SERIES_2:13
theorem Th14: :: SERIES_2:14
theorem Th15: :: SERIES_2:15
theorem Th16: :: SERIES_2:16
theorem Th17: :: SERIES_2:17
theorem Th18: :: SERIES_2:18
Lemma43:
for n being Element of NAT holds ((n |^ 2) * (((2 * (n |^ 2)) + (2 * n)) - 1)) + (((n + 1) |^ 3) * 12) = ((n + 2) |^ 2) * (((2 * ((n + 1) |^ 2)) + (2 * (n + 1))) - 1)
theorem Th19: :: SERIES_2:19
Lemma44:
for n being Element of NAT holds ((n + 2) * ((2 * (n + 1)) + 1)) * ((((3 * ((n + 1) |^ 4)) + (6 * ((n + 1) |^ 3))) - (3 * (n + 1))) + 1) = ((((((6 * (n |^ 6)) + (57 * (n |^ 5))) + (216 * (n |^ 4))) + (414 * (n |^ 3))) + (419 * (n |^ 2))) + (211 * n)) + 42
theorem Th20: :: SERIES_2:20
Lemma45:
for n being Element of NAT holds ((n + 2) |^ 2) * (((((3 * ((n + 1) |^ 4)) + (6 * ((n + 1) |^ 3))) - ((n + 1) |^ 2)) - (4 * (n + 1))) + 2) = ((n |^ 2) * (((((3 * (n |^ 4)) + (6 * (n |^ 3))) - (n |^ 2)) - (4 * n)) + 2)) + (((n + 1) |^ 5) * 24)
Lemma46:
for n being Element of NAT holds (2 |^ (n + 2)) + ((- 1) |^ (n + 2)) > 0
Lemma47:
for n being Element of NAT holds (2 |^ (n + 3)) + ((- 1) |^ (n + 3)) > 0
theorem Th21: :: SERIES_2:21
theorem Th22: :: SERIES_2:22
theorem Th23: :: SERIES_2:23
theorem Th24: :: SERIES_2:24
theorem Th25: :: SERIES_2:25
theorem Th26: :: SERIES_2:26
theorem Th27: :: SERIES_2:27
theorem Th28: :: SERIES_2:28
theorem Th29: :: SERIES_2:29
theorem Th30: :: SERIES_2:30
theorem Th31: :: SERIES_2:31
theorem Th32: :: SERIES_2:32
theorem Th33: :: SERIES_2:33
theorem Th34: :: SERIES_2:34
theorem Th35: :: SERIES_2:35
theorem Th36: :: SERIES_2:36
theorem Th37: :: SERIES_2:37
theorem Th38: :: SERIES_2:38
theorem Th39: :: SERIES_2:39
theorem Th40: :: SERIES_2:40
theorem Th41: :: SERIES_2:41
theorem Th42: :: SERIES_2:42