:: TAYLOR_2 semantic presentation
theorem Th1: :: TAYLOR_2:1
:: deftheorem Def1 defines Maclaurin TAYLOR_2:def 1 :
theorem Th2: :: TAYLOR_2:2
theorem Th3: :: TAYLOR_2:3
for b
1 being
Natfor b
2 being
PartFunc of
REAL ,
REAL for b
3, b
4 being
Real holds
( 0
< b
4 & b
2 is_differentiable_on b
1 + 1,
].(b3 - b4),(b3 + b4).[ implies for b
5 being
Real holds
not ( b
5 in ].(b3 - b4),(b3 + b4).[ & ( for b
6 being
Real holds
not ( 0
< b
6 & b
6 < 1 &
abs ((b2 . b5) - ((Partial_Sums (Taylor b2,].(b3 - b4),(b3 + b4).[,b3,b5)) . b1)) = abs (((((diff b2,].(b3 - b4),(b3 + b4).[) . (b1 + 1)) . (b3 + (b6 * (b5 - b3)))) * ((b5 - b3) |^ (b1 + 1))) / ((b1 + 1) ! )) ) ) ) )
theorem Th4: :: TAYLOR_2:4
theorem Th5: :: TAYLOR_2:5
theorem Th6: :: TAYLOR_2:6
theorem Th7: :: TAYLOR_2:7
theorem Th8: :: TAYLOR_2:8
theorem Th9: :: TAYLOR_2:9
theorem Th10: :: TAYLOR_2:10
theorem Th11: :: TAYLOR_2:11
for b
1 being
Real holds
not ( 0
< b
1 & ( for b
2, b
3 being
Real holds
not ( 0
<= b
2 & 0
<= b
3 & ( for b
4 being
Natfor b
5, b
6 being
Real holds
( b
5 in ].(- b1),b1.[ & 0
< b
6 & b
6 < 1 implies
abs (((((diff exp_R ,].(- b1),b1.[) . b4) . (b6 * b5)) * (b5 |^ b4)) / (b4 ! )) <= (b2 * (b3 |^ b4)) / (b4 ! ) ) ) ) ) )
theorem Th12: :: TAYLOR_2:12
for b
1, b
2 being
Real holds
( b
1 >= 0 & b
2 >= 0 implies for b
3 being
Real holds
not ( b
3 > 0 & ( for b
4 being
Nat holds
ex b
5 being
Nat st
( b
4 <= b
5 & not
(b1 * (b2 |^ b5)) / (b5 ! ) < b
3 ) ) ) )
theorem Th13: :: TAYLOR_2:13
for b
1, b
2 being
Real holds
not ( 0
< b
1 & 0
< b
2 & ( for b
3 being
Nat holds
ex b
4 being
Nat st
( b
3 <= b
4 & ex b
5, b
6 being
Real st
( b
5 in ].(- b1),b1.[ & 0
< b
6 & b
6 < 1 & not
abs (((((diff exp_R ,].(- b1),b1.[) . b4) . (b6 * b5)) * (b5 |^ b4)) / (b4 ! )) < b
2 ) ) ) )
theorem Th14: :: TAYLOR_2:14
theorem Th15: :: TAYLOR_2:15
theorem Th16: :: TAYLOR_2:16
for b
1, b
2 being
Real holds
( 0
< b
1 implies (
Maclaurin exp_R ,
].(- b1),b1.[,b
2 = b
2 ExpSeq &
Maclaurin exp_R ,
].(- b1),b1.[,b
2 is
absolutely_summable &
exp_R . b
2 = Sum (Maclaurin exp_R ,].(- b1),b1.[,b2) ) )
theorem Th17: :: TAYLOR_2:17
theorem Th18: :: TAYLOR_2:18
theorem Th19: :: TAYLOR_2:19
for b
1 being
Realfor b
2 being
Nat holds
(
(diff sin ,].(- b1),b1.[) . (2 * b2) = ((- 1) |^ b2) (#) (sin | ].(- b1),b1.[) &
(diff sin ,].(- b1),b1.[) . ((2 * b2) + 1) = ((- 1) |^ b2) (#) (cos | ].(- b1),b1.[) &
(diff cos ,].(- b1),b1.[) . (2 * b2) = ((- 1) |^ b2) (#) (cos | ].(- b1),b1.[) &
(diff cos ,].(- b1),b1.[) . ((2 * b2) + 1) = ((- 1) |^ (b2 + 1)) (#) (sin | ].(- b1),b1.[) )
theorem Th20: :: TAYLOR_2:20
for b
1 being
Natfor b
2, b
3 being
Real holds
( b
2 > 0 implies (
(Maclaurin sin ,].(- b2),b2.[,b3) . (2 * b1) = 0 &
(Maclaurin sin ,].(- b2),b2.[,b3) . ((2 * b1) + 1) = (((- 1) |^ b1) * (b3 |^ ((2 * b1) + 1))) / (((2 * b1) + 1) ! ) &
(Maclaurin cos ,].(- b2),b2.[,b3) . (2 * b1) = (((- 1) |^ b1) * (b3 |^ (2 * b1))) / ((2 * b1) ! ) &
(Maclaurin cos ,].(- b2),b2.[,b3) . ((2 * b1) + 1) = 0 ) )
theorem Th21: :: TAYLOR_2:21
theorem Th22: :: TAYLOR_2:22
theorem Th23: :: TAYLOR_2:23
theorem Th24: :: TAYLOR_2:24
theorem Th25: :: TAYLOR_2:25
theorem Th26: :: TAYLOR_2:26
theorem Th27: :: TAYLOR_2:27
theorem Th28: :: TAYLOR_2:28
for b
1, b
2 being
Real holds
( b
1 > 0 implies (
Partial_Sums (Maclaurin sin ,].(- b1),b1.[,b2) is
convergent &
sin . b
2 = Sum (Maclaurin sin ,].(- b1),b1.[,b2) &
Partial_Sums (Maclaurin cos ,].(- b1),b1.[,b2) is
convergent &
cos . b
2 = Sum (Maclaurin cos ,].(- b1),b1.[,b2) ) )