Article TAYLOR_2, MML version 4.99.1005

:: TAYLOR_2:th 1
theorem
for b1 being Element of REAL
for b2 being Element of NAT holds
   abs (b1 |^ b2) = (abs b1) |^ b2;

:: TAYLOR_2:funcnot 1 => TAYLOR_2:func 1
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Element of bool REAL;
  let a3 be real set;
  func Maclaurin(A1,A2,A3) -> Function-like quasi_total Relation of NAT,REAL equals
    Taylor(a1,a2,0,a3);
end;

:: TAYLOR_2:def 1
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of bool REAL
for b3 being real set holds
   Maclaurin(b1,b2,b3) = Taylor(b1,b2,0,b3);

:: TAYLOR_2:th 2
theorem
for b1 being Element of NAT
for b2 being Function-like Relation of REAL,REAL
for b3 being Element of REAL
   st 0 < b3 &
      b2 is_differentiable_on b1 + 1,].- b3,b3.[
for b4 being Element of REAL
      st b4 in ].- b3,b3.[
   holds ex b5 being Element of REAL st
      0 < b5 &
       b5 < 1 &
       b2 . b4 = ((Partial_Sums Maclaurin(b2,].- b3,b3.[,b4)) . b1) + (((((diff(b2,].- b3,b3.[)) . (b1 + 1)) . (b5 * b4)) * (b4 |^ (b1 + 1))) / ((b1 + 1) !));

:: TAYLOR_2:th 3
theorem
for b1 being Element of NAT
for b2 being Function-like Relation of REAL,REAL
for b3, b4 being Element of REAL
   st 0 < b4 &
      b2 is_differentiable_on b1 + 1,].b3 - b4,b3 + b4.[
for b5 being Element of REAL
      st b5 in ].b3 - b4,b3 + b4.[
   holds ex b6 being Element of REAL st
      0 < b6 &
       b6 < 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) !));

:: TAYLOR_2:th 4
theorem
for b1 being Element of NAT
for b2 being Function-like Relation of REAL,REAL
for b3 being Element of REAL
   st 0 < b3 &
      b2 is_differentiable_on b1 + 1,].- b3,b3.[
for b4 being Element of REAL
      st b4 in ].- b3,b3.[
   holds ex b5 being Element of REAL st
      0 < b5 &
       b5 < 1 &
       abs ((b2 . b4) - ((Partial_Sums Maclaurin(b2,].- b3,b3.[,b4)) . b1)) = abs (((((diff(b2,].- b3,b3.[)) . (b1 + 1)) . (b5 * b4)) * (b4 |^ (b1 + 1))) / ((b1 + 1) !));

:: TAYLOR_2:th 5
theorem
for b1 being open Element of bool REAL holds
   exp_R `| b1 = exp_R | b1 & proj1 (exp_R | b1) = b1;

:: TAYLOR_2:th 6
theorem
for b1 being open Element of bool REAL
for b2 being Element of NAT holds
   (diff(exp_R,b1)) . b2 = exp_R | b1;

:: TAYLOR_2:th 7
theorem
for b1 being open Element of bool REAL
for b2 being Element of NAT
for b3 being Element of REAL
      st b3 in b1
   holds ((diff(exp_R,b1)) . b2) . b3 = exp_R . b3;

:: TAYLOR_2:th 8
theorem
for b1 being Element of NAT
for b2, b3 being Element of REAL
      st 0 < b2
   holds (Maclaurin(exp_R,].- b2,b2.[,b3)) . b1 = (b3 |^ b1) / (b1 !);

:: TAYLOR_2:th 9
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of REAL
      st b3 in ].- b2,b2.[ & 0 < b4 & b4 < 1
   holds abs (((((diff(exp_R,].- b2,b2.[)) . (b1 + 1)) . (b4 * b3)) * (b3 |^ (b1 + 1))) / ((b1 + 1) !)) <= ((abs (exp_R . (b4 * b3))) * ((abs b3) |^ (b1 + 1))) / ((b1 + 1) !);

:: TAYLOR_2:th 10
theorem
for b1 being open Element of bool REAL
for b2 being Element of NAT holds
   exp_R is_differentiable_on b2,b1;

:: TAYLOR_2:th 11
theorem
for b1 being Element of REAL
      st 0 < b1
   holds ex b2, b3 being Element of REAL st
      0 <= b2 &
       0 <= b3 &
       (for b4 being Element of NAT
       for b5, b6 being Element of REAL
             st b5 in ].- b1,b1.[ & 0 < b6 & b6 < 1
          holds abs (((((diff(exp_R,].- b1,b1.[)) . b4) . (b6 * b5)) * (b5 |^ b4)) / (b4 !)) <= (b2 * (b3 |^ b4)) / (b4 !));

:: TAYLOR_2:th 12
theorem
for b1, b2 being Element of REAL
   st 0 <= b1 & 0 <= b2
for b3 being Element of REAL
      st 0 < b3
   holds ex b4 being Element of NAT st
      for b5 being Element of NAT
            st b4 <= b5
         holds (b1 * (b2 |^ b5)) / (b5 !) < b3;

:: TAYLOR_2:th 13
theorem
for b1, b2 being Element of REAL
      st 0 < b1 & 0 < b2
   holds ex b3 being Element of NAT st
      for b4 being Element of NAT
         st b3 <= b4
      for b5, b6 being Element of REAL
            st b5 in ].- b1,b1.[ & 0 < b6 & b6 < 1
         holds abs (((((diff(exp_R,].- b1,b1.[)) . b4) . (b6 * b5)) * (b5 |^ b4)) / (b4 !)) < b2;

:: TAYLOR_2:th 14
theorem
for b1, b2 being Element of REAL
      st 0 < b1 & 0 < b2
   holds ex b3 being Element of NAT st
      for b4 being Element of NAT
         st b3 <= b4
      for b5 being real set
            st b5 in ].- b1,b1.[
         holds abs ((exp_R . b5) - ((Partial_Sums Maclaurin(exp_R,].- b1,b1.[,b5)) . b4)) < b2;

:: TAYLOR_2:th 15
theorem
for b1 being Element of REAL holds
   b1 ExpSeq is absolutely_summable;

:: TAYLOR_2:th 16
theorem
for b1, b2 being Element of REAL
      st 0 < b1
   holds Maclaurin(exp_R,].- b1,b1.[,b2) = b2 ExpSeq &
    Maclaurin(exp_R,].- b1,b1.[,b2) is absolutely_summable &
    exp_R . b2 = Sum Maclaurin(exp_R,].- b1,b1.[,b2);

:: TAYLOR_2:th 17
theorem
for b1 being open Element of bool REAL holds
   sin `| b1 = cos | b1 &
    cos `| b1 = (- sin) | b1 &
    proj1 (sin | b1) = b1 &
    proj1 (cos | b1) = b1;

:: TAYLOR_2:th 18
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of bool REAL
      st b1 is_differentiable_on b2
   holds (- b1) `| b2 = - (b1 `| b2);

:: TAYLOR_2:th 19
theorem
for b1 being open Element of bool REAL
for b2 being Element of NAT holds
   (diff(sin,b1)) . (2 * b2) = ((- 1) |^ b2) (#) (sin | b1) &
    (diff(sin,b1)) . ((2 * b2) + 1) = ((- 1) |^ b2) (#) (cos | b1) &
    (diff(cos,b1)) . (2 * b2) = ((- 1) |^ b2) (#) (cos | b1) &
    (diff(cos,b1)) . ((2 * b2) + 1) = ((- 1) |^ (b2 + 1)) (#) (sin | b1);

:: TAYLOR_2:th 20
theorem
for b1 being Element of NAT
for b2, b3 being Element of REAL
      st 0 < b2
   holds (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;

:: TAYLOR_2:th 21
theorem
for b1 being open Element of bool REAL
for b2 being Element of NAT holds
   sin is_differentiable_on b2,b1 & cos is_differentiable_on b2,b1;

:: TAYLOR_2:th 22
theorem
for b1 being Element of REAL
      st 0 < b1
   holds ex b2, b3 being Element of REAL st
      0 <= b2 &
       0 <= b3 &
       (for b4 being Element of NAT
       for b5, b6 being Element of REAL
             st b5 in ].- b1,b1.[ & 0 < b6 & b6 < 1
          holds abs (((((diff(sin,].- b1,b1.[)) . b4) . (b6 * b5)) * (b5 |^ b4)) / (b4 !)) <= (b2 * (b3 |^ b4)) / (b4 !) &
           abs (((((diff(cos,].- b1,b1.[)) . b4) . (b6 * b5)) * (b5 |^ b4)) / (b4 !)) <= (b2 * (b3 |^ b4)) / (b4 !));

:: TAYLOR_2:th 23
theorem
for b1, b2 being Element of REAL
      st 0 < b1 & 0 < b2
   holds ex b3 being Element of NAT st
      for b4 being Element of NAT
         st b3 <= b4
      for b5, b6 being Element of REAL
            st b5 in ].- b1,b1.[ & 0 < b6 & b6 < 1
         holds abs (((((diff(sin,].- b1,b1.[)) . b4) . (b6 * b5)) * (b5 |^ b4)) / (b4 !)) < b2 &
          abs (((((diff(cos,].- b1,b1.[)) . b4) . (b6 * b5)) * (b5 |^ b4)) / (b4 !)) < b2;

:: TAYLOR_2:th 24
theorem
for b1, b2 being Element of REAL
      st 0 < b1 & 0 < b2
   holds ex b3 being Element of NAT st
      for b4 being Element of NAT
         st b3 <= b4
      for b5 being real set
            st b5 in ].- b1,b1.[
         holds abs ((sin . b5) - ((Partial_Sums Maclaurin(sin,].- b1,b1.[,b5)) . b4)) < b2 &
          abs ((cos . b5) - ((Partial_Sums Maclaurin(cos,].- b1,b1.[,b5)) . b4)) < b2;

:: TAYLOR_2:th 25
theorem
for b1, b2 being Element of REAL
for b3 being Element of NAT
      st 0 < b1
   holds (Partial_Sums Maclaurin(sin,].- b1,b1.[,b2)) . ((2 * b3) + 1) = (Partial_Sums (b2 P_sin)) . b3 &
    (Partial_Sums Maclaurin(cos,].- b1,b1.[,b2)) . ((2 * b3) + 1) = (Partial_Sums (b2 P_cos)) . b3;

:: TAYLOR_2:th 26
theorem
for b1, b2 being Element of REAL
for b3 being Element of NAT
      st 0 < b1 & 0 < b3
   holds (Partial_Sums Maclaurin(sin,].- b1,b1.[,b2)) . (2 * b3) = (Partial_Sums (b2 P_sin)) . (b3 - 1) &
    (Partial_Sums Maclaurin(cos,].- b1,b1.[,b2)) . (2 * b3) = (Partial_Sums (b2 P_cos)) . b3;

:: TAYLOR_2:th 27
theorem
for b1, b2 being Element of REAL
for b3 being Element of NAT
      st 0 < b1
   holds (Partial_Sums Maclaurin(cos,].- b1,b1.[,b2)) . (2 * b3) = (Partial_Sums (b2 P_cos)) . b3;

:: TAYLOR_2:th 28
theorem
for b1, b2 being Element of REAL
      st 0 < b1
   holds Partial_Sums Maclaurin(sin,].- b1,b1.[,b2) is convergent &
    sin . b2 = Sum Maclaurin(sin,].- b1,b1.[,b2) &
    Partial_Sums Maclaurin(cos,].- b1,b1.[,b2) is convergent &
    cos . b2 = Sum Maclaurin(cos,].- b1,b1.[,b2);