Article SERIES_4, MML version 4.99.1005
:: SERIES_4:th 1
theorem
for b1, b2, b3 being real set holds
((b1 + b2) + b3) |^ 2 = (((((b1 |^ 2) + (b2 |^ 2)) + (b3 |^ 2)) + ((2 * b1) * b2)) + ((2 * b1) * b3)) + ((2 * b2) * b3);
:: SERIES_4:th 2
theorem
for b1, b2 being real set holds
(b1 + b2) |^ 3 = (((b1 |^ 3) + ((3 * (b1 |^ 2)) * b2)) + ((3 * (b2 |^ 2)) * b1)) + (b2 |^ 3);
:: SERIES_4:th 3
theorem
for b1, b2, b3 being real set holds
((b1 - b2) + b3) |^ 2 = (((((b1 |^ 2) + (b2 |^ 2)) + (b3 |^ 2)) - ((2 * b1) * b2)) + ((2 * b1) * b3)) - ((2 * b2) * b3);
:: SERIES_4:th 4
theorem
for b1, b2, b3 being real set holds
((b1 - b2) - b3) |^ 2 = (((((b1 |^ 2) + (b2 |^ 2)) + (b3 |^ 2)) - ((2 * b1) * b2)) - ((2 * b1) * b3)) + ((2 * b2) * b3);
:: SERIES_4:th 5
theorem
for b1, b2 being real set holds
(b1 - b2) |^ 3 = (((b1 |^ 3) - ((3 * (b1 |^ 2)) * b2)) + ((3 * (b2 |^ 2)) * b1)) - (b2 |^ 3);
:: SERIES_4:th 6
theorem
for b1, b2 being real set holds
(b1 + b2) |^ 4 = ((((b1 |^ 4) + ((4 * (b1 |^ 3)) * b2)) + ((6 * (b1 |^ 2)) * (b2 |^ 2))) + ((4 * (b2 |^ 3)) * b1)) + (b2 |^ 4);
:: SERIES_4:th 7
theorem
for b1, b2, b3, b4 being real set holds
(((b1 + b2) + b3) + b4) |^ 2 = ((((((b1 |^ 2) + (b2 |^ 2)) + (b3 |^ 2)) + (b4 |^ 2)) + ((((2 * b1) * b2) + ((2 * b1) * b3)) + ((2 * b1) * b4))) + (((2 * b2) * b3) + ((2 * b2) * b4))) + ((2 * b3) * b4);
:: SERIES_4:th 8
theorem
for b1, b2, b3 being real set holds
((b1 + b2) + b3) |^ 3 = ((((((b1 |^ 3) + (b2 |^ 3)) + (b3 |^ 3)) + (((3 * (b1 |^ 2)) * b2) + ((3 * (b1 |^ 2)) * b3))) + (((3 * (b2 |^ 2)) * b1) + ((3 * (b2 |^ 2)) * b3))) + (((3 * (b3 |^ 2)) * b1) + ((3 * (b3 |^ 2)) * b2))) + (((6 * b1) * b2) * b3);
:: SERIES_4:th 9
theorem
for b1 being Element of NAT
for b2 being real set
st b2 <> 0
holds (((1 / b2) |^ (b1 + 1)) + (b2 |^ (b1 + 1))) |^ 2 = (((1 / b2) |^ ((2 * b1) + 2)) + (b2 |^ ((2 * b1) + 2))) + 2;
:: SERIES_4:th 10
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Function-like quasi_total Relation of NAT,REAL
st b2 <> 1 &
(for b4 being Element of NAT holds
b3 . b4 = b2 |^ b4)
holds (Partial_Sums b3) . b1 = (1 - (b2 |^ (b1 + 1))) / (1 - b2);
:: SERIES_4:th 11
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
st b1 <> 1 &
b1 <> 0 &
(for b3 being Element of NAT holds
b2 . b3 = (1 / b1) |^ b3)
for b3 being Element of NAT holds
(Partial_Sums b2) . b3 = (((1 / b1) |^ b3) - b1) / (1 - b1);
:: SERIES_4:th 12
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
st for b3 being Element of NAT holds
b2 . b3 = ((10 |^ b3) + (2 * b3)) + 1
holds (Partial_Sums b2) . b1 = (((10 |^ (b1 + 1)) / 9) - (1 / 9)) + ((b1 + 1) |^ 2);
:: SERIES_4:th 13
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
st for b3 being Element of NAT holds
b2 . b3 = ((2 * b3) - 1) + ((1 / 2) |^ b3)
holds (Partial_Sums b2) . b1 = ((b1 |^ 2) + 1) - ((1 / 2) |^ b1);
:: SERIES_4:th 14
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
st for b3 being Element of NAT holds
b2 . b3 = b3 * ((1 / 2) |^ b3)
holds (Partial_Sums b2) . b1 = 2 - ((2 + b1) * ((1 / 2) |^ b1));
:: SERIES_4:th 15
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st for b2 being Element of NAT holds
b1 . b2 = (((1 / 2) |^ b2) + (2 |^ b2)) |^ 2
for b2 being Element of NAT holds
(Partial_Sums b1) . b2 = (((- (((1 / 4) |^ b2) / 3)) + ((4 |^ (b2 + 1)) / 3)) + (2 * b2)) + 3;
:: SERIES_4:th 16
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st for b2 being Element of NAT holds
b1 . b2 = (((1 / 3) |^ b2) + (3 |^ b2)) |^ 2
for b2 being Element of NAT holds
(Partial_Sums b1) . b2 = (((- (((1 / 9) |^ b2) / 8)) + ((9 |^ (b2 + 1)) / 8)) + (2 * b2)) + 3;
:: SERIES_4: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 * (2 |^ b2)
for b2 being Element of NAT holds
(Partial_Sums b1) . b2 = ((b2 * (2 |^ (b2 + 1))) - (2 |^ (b2 + 1))) + 2;
:: SERIES_4:th 18
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) * (3 |^ b2)
for b2 being Element of NAT holds
(Partial_Sums b1) . b2 = (b2 * (3 |^ (b2 + 1))) + 1;
:: SERIES_4:th 19
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
st b1 <> 1 &
(for b3 being Element of NAT holds
b2 . b3 = b3 * (b1 |^ b3))
for b3 being Element of NAT holds
(Partial_Sums b2) . b3 = ((b1 * (1 - (b1 |^ b3))) / ((1 - b1) |^ 2)) - ((b3 * (b1 |^ (b3 + 1))) / (1 - b1));
:: SERIES_4:th 20
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
st for b3 being Element of NAT holds
b2 . b3 = 1 / ((2 -Root (b3 + 1)) + (2 -Root b3))
holds (Partial_Sums b2) . b1 = 2 -Root (b1 + 1);
:: SERIES_4:th 21
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 / 2) |^ b2)
for b2 being Element of NAT holds
(Partial_Sums b1) . b2 = ((2 |^ (b2 + 1)) - ((1 / 2) |^ b2)) + 1;
:: SERIES_4: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) + (b2 / ((b2 + 1) !))
for b2 being Element of NAT
st 1 <= b2
holds (Partial_Sums b1) . b2 = (b2 + 1) ! - (1 / ((b2 + 1) !));
:: SERIES_4:th 23
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
st b1 <> 1 &
(for b3 being Element of NAT
st 1 <= b3
holds b2 . b3 = (b1 / (b1 - 1)) |^ b3 &
b2 . 0 = 0)
for b3 being Element of NAT
st 1 <= b3
holds (Partial_Sums b2) . b3 = b1 * (((b1 / (b1 - 1)) |^ b3) - 1);
:: SERIES_4:th 24
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) * (((3 * b2) - 1) / 4) &
b1 . 0 = 0
for b2 being Element of NAT
st 1 <= b2
holds (Partial_Sums b1) . b2 = ((2 |^ b2) * (((3 * b2) - 4) / 2)) + 2;
:: SERIES_4:th 25
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
st for b3 being Element of NAT holds
b2 . b3 = (b3 + 1) / (b3 + 2)
holds (Partial_Product b2) . b1 = 1 / (b1 + 2);
:: SERIES_4:th 26
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
st for b3 being Element of NAT holds
b2 . b3 = 1 / (b3 + 1)
holds (Partial_Product b2) . b1 = 1 / ((b1 + 1) !);
:: SERIES_4: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 = b2 & b1 . 0 = 1
for b2 being Element of NAT
st 1 <= b2
holds (Partial_Product b1) . b2 = b2 !;
:: SERIES_4:th 28
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
st for b3 being Element of NAT
st 1 <= b3
holds b2 . b3 = b1 / b3 & b2 . 0 = 1
for b3 being Element of NAT
st 1 <= b3
holds (Partial_Product b2) . b3 = (b1 |^ b3) / (b3 !);
:: SERIES_4:th 29
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
st for b3 being Element of NAT
st 1 <= b3
holds b2 . b3 = b1 & b2 . 0 = 1
for b3 being Element of NAT
st 1 <= b3
holds (Partial_Product b2) . b3 = b1 |^ b3;
:: SERIES_4:th 30
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st for b2 being Element of NAT
st 2 <= b2
holds b1 . b2 = 1 - (1 / (b2 |^ 2)) &
b1 . 0 = 1 &
b1 . 1 = 1
for b2 being Element of NAT
st 2 <= b2
holds (Partial_Product b1) . b2 = (b2 + 1) / (2 * b2);