:: 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
n being
Element of
NAT for
f being
PartFunc of
REAL ,
REAL for
x0,
r being
Real st 0
< r &
f is_differentiable_on n + 1,
].(x0 - r),(x0 + r).[ holds
for
x being
Real st
x in ].(x0 - r),(x0 + r).[ holds
ex
s being
Real st
( 0
< s &
s < 1 &
abs ((f . x) - ((Partial_Sums (Taylor f,].(x0 - r),(x0 + r).[,x0,x)) . n)) = abs (((((diff f,].(x0 - r),(x0 + r).[) . (n + 1)) . (x0 + (s * (x - x0)))) * ((x - x0) |^ (n + 1))) / ((n + 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
theorem Th12: :: TAYLOR_2:12
theorem Th13: :: TAYLOR_2:13
theorem Th14: :: TAYLOR_2:14
theorem Th15: :: TAYLOR_2:15
theorem Th16: :: TAYLOR_2:16
for
r,
x being
Real st 0
< r holds
(
Maclaurin exp_R ,
].(- r),r.[,
x = x ExpSeq &
Maclaurin exp_R ,
].(- r),r.[,
x is
absolutely_summable &
exp_R . x = Sum (Maclaurin exp_R ,].(- r),r.[,x) )
theorem Th17: :: TAYLOR_2:17
theorem Th18: :: TAYLOR_2:18
theorem Th19: :: TAYLOR_2:19
for
r being
Real for
n being
Element of
NAT holds
(
(diff sin ,].(- r),r.[) . (2 * n) = ((- 1) |^ n) (#) (sin | ].(- r),r.[) &
(diff sin ,].(- r),r.[) . ((2 * n) + 1) = ((- 1) |^ n) (#) (cos | ].(- r),r.[) &
(diff cos ,].(- r),r.[) . (2 * n) = ((- 1) |^ n) (#) (cos | ].(- r),r.[) &
(diff cos ,].(- r),r.[) . ((2 * n) + 1) = ((- 1) |^ (n + 1)) (#) (sin | ].(- r),r.[) )
theorem Th20: :: TAYLOR_2:20
for
n being
Element of
NAT for
r,
x being
Real st
r > 0 holds
(
(Maclaurin sin ,].(- r),r.[,x) . (2 * n) = 0 &
(Maclaurin sin ,].(- r),r.[,x) . ((2 * n) + 1) = (((- 1) |^ n) * (x |^ ((2 * n) + 1))) / (((2 * n) + 1) ! ) &
(Maclaurin cos ,].(- r),r.[,x) . (2 * n) = (((- 1) |^ n) * (x |^ (2 * n))) / ((2 * n) ! ) &
(Maclaurin cos ,].(- r),r.[,x) . ((2 * n) + 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
r,
x being
Real st
r > 0 holds
(
Partial_Sums (Maclaurin sin ,].(- r),r.[,x) is
convergent &
sin . x = Sum (Maclaurin sin ,].(- r),r.[,x) &
Partial_Sums (Maclaurin cos ,].(- r),r.[,x) is
convergent &
cos . x = Sum (Maclaurin cos ,].(- r),r.[,x) )