Article SERIES_1, MML version 4.99.1005
:: SERIES_1:th 1
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
st 0 < b1 &
b1 < 1 &
(for b3 being Element of NAT holds
b2 . b3 = b1 to_power (b3 + 1))
holds b2 is convergent & lim b2 = 0;
:: SERIES_1:th 2
theorem
for b1 being Element of NAT
for b2 being real set holds
(abs b2) to_power b1 = abs (b2 to_power b1);
:: SERIES_1:th 3
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
st abs b1 < 1 &
(for b3 being Element of NAT holds
b2 . b3 = b1 to_power (b3 + 1))
holds b2 is convergent & lim b2 = 0;
:: SERIES_1:funcnot 1 => SERIES_1:func 1
definition
let a1 be Function-like quasi_total Relation of NAT,REAL;
func Partial_Sums A1 -> Function-like quasi_total Relation of NAT,REAL means
it . 0 = a1 . 0 &
(for b1 being Element of NAT holds
it . (b1 + 1) = (it . b1) + (a1 . (b1 + 1)));
end;
:: SERIES_1:def 1
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
b2 = Partial_Sums b1
iff
b2 . 0 = b1 . 0 &
(for b3 being Element of NAT holds
b2 . (b3 + 1) = (b2 . b3) + (b1 . (b3 + 1)));
:: SERIES_1:attrnot 1 => SERIES_1:attr 1
definition
let a1 be Function-like quasi_total Relation of NAT,REAL;
attr a1 is summable means
Partial_Sums a1 is convergent;
end;
:: SERIES_1:dfs 2
definiens
let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
a1 is summable
it is sufficient to prove
thus Partial_Sums a1 is convergent;
:: SERIES_1:def 2
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is summable
iff
Partial_Sums b1 is convergent;
:: SERIES_1:funcnot 2 => SERIES_1:func 2
definition
let a1 be Function-like quasi_total Relation of NAT,REAL;
func Sum A1 -> Element of REAL equals
lim Partial_Sums a1;
end;
:: SERIES_1:def 3
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
Sum b1 = lim Partial_Sums b1;
:: SERIES_1:th 7
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is summable
holds b1 is convergent & lim b1 = 0;
:: SERIES_1:th 8
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
(Partial_Sums b1) + Partial_Sums b2 = Partial_Sums (b1 + b2);
:: SERIES_1:th 9
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
(Partial_Sums b1) - Partial_Sums b2 = Partial_Sums (b1 - b2);
:: SERIES_1:th 10
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is summable & b2 is summable
holds b1 + b2 is summable &
Sum (b1 + b2) = (Sum b1) + Sum b2;
:: SERIES_1:th 11
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is summable & b2 is summable
holds b1 - b2 is summable &
Sum (b1 - b2) = (Sum b1) - Sum b2;
:: SERIES_1:th 12
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
Partial_Sums (b1 (#) b2) = b1 (#) Partial_Sums b2;
:: SERIES_1:th 13
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
st b2 is summable
holds b1 (#) b2 is summable & Sum (b1 (#) b2) = b1 * Sum b2;
:: SERIES_1:th 14
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st for b3 being Element of NAT holds
b2 . b3 = b1 . 0
holds Partial_Sums (b1 ^\ 1) = ((Partial_Sums b1) ^\ 1) - b2;
:: SERIES_1:th 15
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is summable
for b2 being Element of NAT holds
b1 ^\ b2 is summable;
:: SERIES_1:th 16
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st ex b2 being Element of NAT st
b1 ^\ b2 is summable
holds b1 is summable;
:: SERIES_1:th 17
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st for b3 being Element of NAT holds
b1 . b3 <= b2 . b3
for b3 being Element of NAT holds
(Partial_Sums b1) . b3 <= (Partial_Sums b2) . b3;
:: SERIES_1:th 18
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is summable
for b2 being Element of NAT holds
Sum b1 = ((Partial_Sums b1) . b2) + Sum (b1 ^\ (b2 + 1));
:: SERIES_1:th 19
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st for b2 being Element of NAT holds
0 <= b1 . b2
holds Partial_Sums b1 is non-decreasing;
:: SERIES_1:th 20
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st for b2 being Element of NAT holds
0 <= b1 . b2
holds Partial_Sums b1 is bounded_above
iff
b1 is summable;
:: SERIES_1:th 21
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is summable &
(for b2 being Element of NAT holds
0 <= b1 . b2)
holds 0 <= Sum b1;
:: SERIES_1:th 22
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st (for b3 being Element of NAT holds
0 <= b1 . b3) &
b2 is summable &
(ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds b1 . b4 <= b2 . b4)
holds b1 is summable;
:: SERIES_1:th 24
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st (for b3 being Element of NAT holds
0 <= b1 . b3 & b1 . b3 <= b2 . b3) &
b2 is summable
holds b1 is summable & Sum b1 <= Sum b2;
:: SERIES_1:th 25
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is summable
iff
for b2 being real set
st 0 < b2
holds ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds abs (((Partial_Sums b1) . b4) - ((Partial_Sums b1) . b3)) < b2;
:: SERIES_1:th 26
theorem
for b1 being Element of NAT
for b2 being real set
st b2 <> 1
holds (Partial_Sums (b2 GeoSeq)) . b1 = (1 - (b2 to_power (b1 + 1))) / (1 - b2);
:: SERIES_1:th 27
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 + 1) = b1 * (b2 . b3))
for b3 being Element of NAT holds
(Partial_Sums b2) . b3 = ((b2 . 0) * (1 - (b1 to_power (b3 + 1)))) / (1 - b1);
:: SERIES_1:th 28
theorem
for b1 being real set
st abs b1 < 1
holds b1 GeoSeq is summable &
Sum (b1 GeoSeq) = 1 / (1 - b1);
:: SERIES_1:th 29
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
st abs b1 < 1 &
(for b3 being Element of NAT holds
b2 . (b3 + 1) = b1 * (b2 . b3))
holds b2 is summable &
Sum b2 = (b2 . 0) / (1 - b1);
:: SERIES_1:th 30
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st (for b3 being Element of NAT holds
0 < b1 . b3 &
b2 . b3 = (b1 . (b3 + 1)) / (b1 . b3)) &
b2 is convergent &
lim b2 < 1
holds b1 is summable;
:: SERIES_1:th 31
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st (for b2 being Element of NAT holds
0 < b1 . b2) &
(ex b2 being Element of NAT st
for b3 being Element of NAT
st b2 <= b3
holds 1 <= (b1 . (b3 + 1)) / (b1 . b3))
holds b1 is not summable;
:: SERIES_1:th 32
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st (for b3 being Element of NAT holds
0 <= b1 . b3 & b2 . b3 = b3 -root (b1 . b3)) &
b2 is convergent &
lim b2 < 1
holds b1 is summable;
:: SERIES_1:th 33
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st (for b3 being Element of NAT holds
0 <= b1 . b3 & b2 . b3 = b3 -root (b1 . b3)) &
(ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds 1 <= b2 . b4)
holds b1 is not summable;
:: SERIES_1:th 34
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st (for b3 being Element of NAT holds
0 <= b1 . b3 & b2 . b3 = b3 -root (b1 . b3)) &
b2 is convergent &
1 < lim b2
holds b1 is not summable;
:: SERIES_1:funcreg 1
registration
let a1, a2 be natural set;
cluster a1 to_power a2 -> natural real;
end;
:: SERIES_1:funcnot 3 => SERIES_1:func 3
definition
let a1, a2 be Element of NAT;
redefine func a1 to_power a2 -> Element of NAT;
end;
:: SERIES_1:th 35
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is non-increasing &
(for b3 being Element of NAT holds
0 <= b1 . b3 &
b2 . b3 = (2 to_power b3) * (b1 . (2 to_power b3)))
holds b1 is summable
iff
b2 is summable;
:: SERIES_1:th 36
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
st 1 < b1 &
(for b3 being Element of NAT
st 1 <= b3
holds b2 . b3 = 1 / (b3 to_power b1))
holds b2 is summable;
:: SERIES_1:th 37
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 = 1 / (b3 to_power b1))
holds b2 is not summable;
:: SERIES_1:attrnot 2 => SERIES_1:attr 2
definition
let a1 be Function-like quasi_total Relation of NAT,REAL;
attr a1 is absolutely_summable means
abs a1 is summable;
end;
:: SERIES_1:dfs 4
definiens
let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
a1 is absolutely_summable
it is sufficient to prove
thus abs a1 is summable;
:: SERIES_1:def 5
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is absolutely_summable
iff
abs b1 is summable;
:: SERIES_1:th 39
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2, b3 being Element of NAT
st b2 <= b3
holds abs (((Partial_Sums b1) . b3) - ((Partial_Sums b1) . b2)) <= abs (((Partial_Sums abs b1) . b3) - ((Partial_Sums abs b1) . b2));
:: SERIES_1:th 40
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is absolutely_summable
holds b1 is summable;
:: SERIES_1:th 41
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st (for b2 being Element of NAT holds
0 <= b1 . b2) &
b1 is summable
holds b1 is absolutely_summable;
:: SERIES_1:th 42
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st (for b3 being Element of NAT holds
b1 . b3 <> 0 &
b2 . b3 = ((abs b1) . (b3 + 1)) / ((abs b1) . b3)) &
b2 is convergent &
lim b2 < 1
holds b1 is absolutely_summable;
:: SERIES_1:th 43
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
st 0 < b1 &
(ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds b1 <= abs (b2 . b4)) &
b2 is convergent
holds lim b2 <> 0;
:: SERIES_1:th 44
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st (for b2 being Element of NAT holds
b1 . b2 <> 0) &
(ex b2 being Element of NAT st
for b3 being Element of NAT
st b2 <= b3
holds 1 <= ((abs b1) . (b3 + 1)) / ((abs b1) . b3))
holds b1 is not summable;
:: SERIES_1:th 45
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st (for b3 being Element of NAT holds
b1 . b3 = b3 -root ((abs b2) . b3)) &
b1 is convergent &
lim b1 < 1
holds b2 is absolutely_summable;
:: SERIES_1:th 46
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st (for b3 being Element of NAT holds
b1 . b3 = b3 -root ((abs b2) . b3)) &
(ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds 1 <= b1 . b4)
holds b2 is not summable;
:: SERIES_1:th 47
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st (for b3 being Element of NAT holds
b1 . b3 = b3 -root ((abs b2) . b3)) &
b1 is convergent &
1 < lim b1
holds b2 is not summable;
:: SERIES_1:funcnot 4 => SERIES_1:func 4
definition
let a1 be Function-like quasi_total Relation of NAT,REAL;
let a2 be natural set;
func Sum(A1,A2) -> Element of REAL equals
(Partial_Sums a1) . a2;
end;
:: SERIES_1:def 6
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being natural set holds
Sum(b1,b2) = (Partial_Sums b1) . b2;
:: SERIES_1:funcnot 5 => SERIES_1:func 5
definition
let a1 be Function-like quasi_total Relation of NAT,REAL;
let a2, a3 be natural set;
func Sum(A1,A2,A3) -> Element of REAL equals
(Sum(a1,a2)) - Sum(a1,a3);
end;
:: SERIES_1:def 7
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2, b3 being natural set holds
Sum(b1,b2,b3) = (Sum(b1,b2)) - Sum(b1,b3);