Article SERIES_2, MML version 4.99.1005

:: SERIES_2:th 1
theorem
for b1 being Element of NAT holds
   abs ((- 1) |^ b1) = 1;

:: SERIES_2:th 2
theorem
for b1 being real set holds
   (b1 + 1) |^ 3 = (((b1 |^ 3) + (3 * (b1 |^ 2))) + (3 * b1)) + 1 &
    (b1 + 1) |^ 4 = ((((b1 |^ 4) + (4 * (b1 |^ 3))) + (6 * (b1 |^ 2))) + (4 * b1)) + 1 &
    (b1 + 1) |^ 5 = (((((b1 |^ 5) + (5 * (b1 |^ 4))) + (10 * (b1 |^ 3))) + (10 * (b1 |^ 2))) + (5 * b1)) + 1;

:: SERIES_2:th 3
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = b2
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 = (b2 * (b2 + 1)) / 2;

:: SERIES_2:th 4
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = 2 * b2
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 = b2 * (b2 + 1);

:: SERIES_2:th 5
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = (2 * b2) + 1
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 = (b2 + 1) |^ 2;

:: SERIES_2:th 6
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = b2 * (b2 + 1)
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 = ((b2 * (b2 + 1)) * (b2 + 2)) / 3;

:: SERIES_2:th 7
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = (b2 * (b2 + 1)) * (b2 + 2)
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 = (((b2 * (b2 + 1)) * (b2 + 2)) * (b2 + 3)) / 4;

:: SERIES_2:th 8
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = ((b2 * (b2 + 1)) * (b2 + 2)) * (b2 + 3)
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 = ((((b2 * (b2 + 1)) * (b2 + 2)) * (b2 + 3)) * (b2 + 4)) / 5;

:: SERIES_2:th 9
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = 1 / (b2 * (b2 + 1))
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 = 1 - (1 / (b2 + 1));

:: SERIES_2:th 10
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = 1 / ((b2 * (b2 + 1)) * (b2 + 2))
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 = (1 / 4) - (1 / ((2 * (b2 + 1)) * (b2 + 2)));

:: SERIES_2:th 11
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = 1 / (((b2 * (b2 + 1)) * (b2 + 2)) * (b2 + 3))
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 = (1 / 18) - (1 / (((3 * (b2 + 1)) * (b2 + 2)) * (b2 + 3)));

:: SERIES_2:th 12
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = b2 |^ 2
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 = ((b2 * (b2 + 1)) * ((2 * b2) + 1)) / 6;

:: SERIES_2:th 13
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = ((- 1) |^ (b2 + 1)) * (b2 |^ 2)
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 = ((((- 1) |^ (b2 + 1)) * b2) * (b2 + 1)) / 2;

:: SERIES_2:th 14
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT
           st 1 <= b2
        holds b1 . b2 = ((2 * b2) - 1) |^ 2 &
         b1 . 0 = 0
for b2 being Element of NAT
      st 1 <= b2
   holds (Partial_Sums b1) . b2 = (b2 * ((4 * (b2 |^ 2)) - 1)) / 3;

:: SERIES_2:th 15
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = b2 |^ 3
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 = ((b2 |^ 2) * ((b2 + 1) |^ 2)) / 4;

:: SERIES_2:th 16
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT
           st 1 <= b2
        holds b1 . b2 = ((2 * b2) - 1) |^ 3 &
         b1 . 0 = 0
for b2 being Element of NAT
      st 1 <= b2
   holds (Partial_Sums b1) . b2 = (b2 |^ 2) * ((2 * (b2 |^ 2)) - 1);

:: SERIES_2:th 17
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = b2 |^ 4
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 = (((b2 * (b2 + 1)) * ((2 * b2) + 1)) * (((3 * (b2 |^ 2)) + (3 * b2)) - 1)) / 30;

:: SERIES_2:th 18
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = ((- 1) |^ (b2 + 1)) * (b2 |^ 4)
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 = (((((- 1) |^ (b2 + 1)) * b2) * (b2 + 1)) * (((b2 |^ 2) + b2) - 1)) / 2;

:: SERIES_2:th 19
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = b2 |^ 5
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 = (((b2 |^ 2) * ((b2 + 1) |^ 2)) * (((2 * (b2 |^ 2)) + (2 * b2)) - 1)) / 12;

:: SERIES_2:th 20
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = b2 |^ 6
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 = (((b2 * (b2 + 1)) * ((2 * b2) + 1)) * ((((3 * (b2 |^ 4)) + (6 * (b2 |^ 3))) - (3 * b2)) + 1)) / 42;

:: SERIES_2:th 21
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = b2 |^ 7
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 = (((b2 |^ 2) * ((b2 + 1) |^ 2)) * (((((3 * (b2 |^ 4)) + (6 * (b2 |^ 3))) - (b2 |^ 2)) - (4 * b2)) + 2)) / 24;

:: SERIES_2:th 22
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = b2 * ((b2 + 1) |^ 2)
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 = (((b2 * (b2 + 1)) * (b2 + 2)) * ((3 * b2) + 5)) / 12;

:: SERIES_2:th 23
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = (b2 * ((b2 + 1) |^ 2)) * (b2 + 2)
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 = ((((b2 * (b2 + 1)) * (b2 + 2)) * (b2 + 3)) * ((2 * b2) + 3)) / 10;

:: SERIES_2:th 24
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = (b2 * (b2 + 1)) * (2 |^ b2)
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 = ((2 |^ (b2 + 1)) * (((b2 |^ 2) - b2) + 2)) - 4;

:: SERIES_2:th 25
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT
           st 1 <= b2
        holds b1 . b2 = 1 / ((b2 - 1) * (b2 + 1)) &
         b1 . 0 = 0
for b2 being Element of NAT
      st 2 <= b2
   holds (Partial_Sums b1) . b2 = ((3 / 4) - (1 / (2 * b2))) - (1 / (2 * (b2 + 1)));

:: SERIES_2:th 26
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT
           st 1 <= b2
        holds b1 . b2 = 1 / (((2 * b2) - 1) * ((2 * b2) + 1)) &
         b1 . 0 = 0
for b2 being Element of NAT
      st 1 <= b2
   holds (Partial_Sums b1) . b2 = b2 / ((2 * b2) + 1);

:: SERIES_2:th 27
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT
           st 1 <= b2
        holds b1 . b2 = 1 / (((3 * b2) - 2) * ((3 * b2) + 1)) &
         b1 . 0 = 0
for b2 being Element of NAT
      st 1 <= b2
   holds (Partial_Sums b1) . b2 = b2 / ((3 * b2) + 1);

:: SERIES_2:th 28
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT
           st 1 <= b2
        holds b1 . b2 = 1 / ((((2 * b2) - 1) * ((2 * b2) + 1)) * ((2 * b2) + 3)) &
         b1 . 0 = 0
for b2 being Element of NAT
      st 1 <= b2
   holds (Partial_Sums b1) . b2 = (1 / 12) - (1 / ((4 * ((2 * b2) + 1)) * ((2 * b2) + 3)));

:: SERIES_2:th 29
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT
           st 1 <= b2
        holds b1 . b2 = 1 / ((((3 * b2) - 2) * ((3 * b2) + 1)) * ((3 * b2) + 4)) &
         b1 . 0 = 0
for b2 being Element of NAT
      st 1 <= b2
   holds (Partial_Sums b1) . b2 = (1 / 24) - (1 / ((6 * ((3 * b2) + 1)) * ((3 * b2) + 4)));

:: SERIES_2:th 30
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = ((2 * b2) - 1) / ((b2 * (b2 + 1)) * (b2 + 2))
for b2 being Element of NAT
      st 1 <= b2
   holds (Partial_Sums b1) . b2 = ((3 / 4) - (2 / (b2 + 2))) + (1 / ((2 * (b2 + 1)) * (b2 + 2)));

:: SERIES_2:th 31
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = (b2 + 2) / ((b2 * (b2 + 1)) * (b2 + 3))
for b2 being Element of NAT
      st 1 <= b2
   holds (Partial_Sums b1) . b2 = (((29 / 36) - (1 / (b2 + 3))) - (3 / ((2 * (b2 + 2)) * (b2 + 3)))) - (4 / (((3 * (b2 + 1)) * (b2 + 2)) * (b2 + 3)));

:: SERIES_2:th 32
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = ((b2 + 1) * (2 |^ b2)) / ((b2 + 2) * (b2 + 3))
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 = ((2 |^ (b2 + 1)) / (b2 + 3)) - (1 / 2);

:: SERIES_2:th 33
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = ((b2 |^ 2) * (4 |^ b2)) / ((b2 + 1) * (b2 + 2))
for b2 being Element of NAT
      st 1 <= b2
   holds (Partial_Sums b1) . b2 = (2 / 3) + (((b2 - 1) * (4 |^ (b2 + 1))) / (3 * (b2 + 2)));

:: SERIES_2:th 34
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = (b2 + 2) / ((b2 * (b2 + 1)) * (2 |^ b2))
for b2 being Element of NAT
      st 1 <= b2
   holds (Partial_Sums b1) . b2 = 1 - (1 / ((b2 + 1) * (2 |^ b2)));

:: SERIES_2:th 35
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = ((2 * b2) + 3) / ((b2 * (b2 + 1)) * (3 |^ b2))
for b2 being Element of NAT
      st 1 <= b2
   holds (Partial_Sums b1) . b2 = 1 - (1 / ((b2 + 1) * (3 |^ b2)));

:: SERIES_2:th 36
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = (((- 1) |^ b2) * (2 |^ (b2 + 1))) / (((2 |^ (b2 + 1)) + ((- 1) |^ (b2 + 1))) * ((2 |^ (b2 + 2)) + ((- 1) |^ (b2 + 2))))
for b2 being Element of NAT holds
   (Partial_Sums b1) . b2 = (1 / 3) + (((- 1) |^ (b2 + 2)) / (3 * ((2 |^ (b2 + 2)) + ((- 1) |^ (b2 + 2)))));

:: SERIES_2:th 37
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = b2 ! * b2
for b2 being Element of NAT
      st 1 <= b2
   holds (Partial_Sums b1) . b2 = (b2 + 1) ! - 1;

:: SERIES_2:th 38
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = b2 / ((b2 + 1) !)
for b2 being Element of NAT
      st 1 <= b2
   holds (Partial_Sums b1) . b2 = 1 - (1 / ((b2 + 1) !));

:: SERIES_2:th 39
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT
           st 1 <= b2
        holds b1 . b2 = (((b2 |^ 2) + b2) - 1) / ((b2 + 2) !) &
         b1 . 0 = 0
for b2 being Element of NAT
      st 1 <= b2
   holds (Partial_Sums b1) . b2 = (1 / 2) - ((b2 + 1) / ((b2 + 2) !));

:: SERIES_2:th 40
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st for b2 being Element of NAT holds
        b1 . b2 = (b2 * (2 |^ b2)) / ((b2 + 2) !)
for b2 being Element of NAT
      st 1 <= b2
   holds (Partial_Sums b1) . b2 = 1 - ((2 |^ (b2 + 1)) / ((b2 + 2) !));

:: SERIES_2:th 41
theorem
for b1, b2 being real set
for b3 being Function-like quasi_total Relation of NAT,REAL
   st for b4 being Element of NAT holds
        b3 . b4 = (b1 * b4) + b2
for b4 being Element of NAT holds
   (Partial_Sums b3) . b4 = ((((b1 * (b4 + 1)) * b4) / 2) + (b4 * b2)) + b2;

:: SERIES_2:th 42
theorem
for b1, b2 being real set
for b3 being Function-like quasi_total Relation of NAT,REAL
   st for b4 being Element of NAT holds
        b3 . b4 = (b1 * b4) + b2
for b4 being Element of NAT holds
   (Partial_Sums b3) . b4 = ((b4 + 1) * ((b3 . 0) + (b3 . b4))) / 2;