:: SERIES_1 semantic presentation

Lemma1: ( 1 / 1 = 1 & 1 / (- 1) = - 1 )
;

theorem Th1: :: SERIES_1:1
for b1 being real number
for b2 being Real_Sequence holds
( 0 < b1 & b1 < 1 & ( for b3 being Nat holds b2 . b3 = b1 to_power (b3 + 1) ) implies ( b2 is convergent & lim b2 = 0 ) )
proof end;

theorem Th2: :: SERIES_1:2
for b1 being Nat
for b2 being real number holds
( b2 <> 0 implies (abs b2) to_power b1 = abs (b2 to_power b1) )
proof end;

theorem Th3: :: SERIES_1:3
for b1 being real number
for b2 being Real_Sequence holds
( abs b1 < 1 & ( for b3 being Nat holds b2 . b3 = b1 to_power (b3 + 1) ) implies ( b2 is convergent & lim b2 = 0 ) )
proof end;

definition
let c1 be Real_Sequence;
func Partial_Sums c1 -> Real_Sequence means :Def1: :: SERIES_1:def 1
( a2 . 0 = a1 . 0 & ( for b1 being Nat holds a2 . (b1 + 1) = (a2 . b1) + (a1 . (b1 + 1)) ) );
existence
ex b1 being Real_Sequence st
( b1 . 0 = c1 . 0 & ( for b2 being Nat holds b1 . (b2 + 1) = (b1 . b2) + (c1 . (b2 + 1)) ) )
proof end;
uniqueness
for b1, b2 being Real_Sequence holds
( b1 . 0 = c1 . 0 & ( for b3 being Nat holds b1 . (b3 + 1) = (b1 . b3) + (c1 . (b3 + 1)) ) & b2 . 0 = c1 . 0 & ( for b3 being Nat holds b2 . (b3 + 1) = (b2 . b3) + (c1 . (b3 + 1)) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def1 defines Partial_Sums SERIES_1:def 1 :
for b1, b2 being Real_Sequence holds
( b2 = Partial_Sums b1 iff ( b2 . 0 = b1 . 0 & ( for b3 being Nat holds b2 . (b3 + 1) = (b2 . b3) + (b1 . (b3 + 1)) ) ) );

definition
let c1 be Real_Sequence;
attr a1 is summable means :Def2: :: SERIES_1:def 2
Partial_Sums a1 is convergent;
func Sum c1 -> Real equals :: SERIES_1:def 3
lim (Partial_Sums a1);
correctness
coherence
lim (Partial_Sums c1) is Real
;
;
end;

:: deftheorem Def2 defines summable SERIES_1:def 2 :
for b1 being Real_Sequence holds
( b1 is summable iff Partial_Sums b1 is convergent );

:: deftheorem Def3 defines Sum SERIES_1:def 3 :
for b1 being Real_Sequence holds Sum b1 = lim (Partial_Sums b1);

theorem Th4: :: SERIES_1:4
canceled;

theorem Th5: :: SERIES_1:5
canceled;

theorem Th6: :: SERIES_1:6
canceled;

theorem Th7: :: SERIES_1:7
for b1 being Real_Sequence holds
( b1 is summable implies ( b1 is convergent & lim b1 = 0 ) )
proof end;

theorem Th8: :: SERIES_1:8
for b1, b2 being Real_Sequence holds (Partial_Sums b1) + (Partial_Sums b2) = Partial_Sums (b1 + b2)
proof end;

theorem Th9: :: SERIES_1:9
for b1, b2 being Real_Sequence holds (Partial_Sums b1) - (Partial_Sums b2) = Partial_Sums (b1 - b2)
proof end;

theorem Th10: :: SERIES_1:10
for b1, b2 being Real_Sequence holds
( b1 is summable & b2 is summable implies ( b1 + b2 is summable & Sum (b1 + b2) = (Sum b1) + (Sum b2) ) )
proof end;

theorem Th11: :: SERIES_1:11
for b1, b2 being Real_Sequence holds
( b1 is summable & b2 is summable implies ( b1 - b2 is summable & Sum (b1 - b2) = (Sum b1) - (Sum b2) ) )
proof end;

theorem Th12: :: SERIES_1:12
for b1 being real number
for b2 being Real_Sequence holds Partial_Sums (b1 (#) b2) = b1 (#) (Partial_Sums b2)
proof end;

theorem Th13: :: SERIES_1:13
for b1 being real number
for b2 being Real_Sequence holds
( b2 is summable implies ( b1 (#) b2 is summable & Sum (b1 (#) b2) = b1 * (Sum b2) ) )
proof end;

theorem Th14: :: SERIES_1:14
for b1, b2 being Real_Sequence holds
( ( for b3 being Nat holds b2 . b3 = b1 . 0 ) implies Partial_Sums (b1 ^\ 1) = ((Partial_Sums b1) ^\ 1) - b2 )
proof end;

theorem Th15: :: SERIES_1:15
for b1 being Real_Sequence holds
( b1 is summable implies for b2 being Nat holds b1 ^\ b2 is summable )
proof end;

theorem Th16: :: SERIES_1:16
for b1 being Real_Sequence holds
( ex b2 being Nat st b1 ^\ b2 is summable implies b1 is summable )
proof end;

theorem Th17: :: SERIES_1:17
for b1, b2 being Real_Sequence holds
( ( for b3 being Nat holds b1 . b3 <= b2 . b3 ) implies for b3 being Nat holds (Partial_Sums b1) . b3 <= (Partial_Sums b2) . b3 )
proof end;

theorem Th18: :: SERIES_1:18
for b1 being Real_Sequence holds
( b1 is summable implies for b2 being Nat holds Sum b1 = ((Partial_Sums b1) . b2) + (Sum (b1 ^\ (b2 + 1))) )
proof end;

theorem Th19: :: SERIES_1:19
for b1 being Real_Sequence holds
( ( for b2 being Nat holds 0 <= b1 . b2 ) implies Partial_Sums b1 is non-decreasing )
proof end;

theorem Th20: :: SERIES_1:20
for b1 being Real_Sequence holds
( ( for b2 being Nat holds 0 <= b1 . b2 ) implies ( Partial_Sums b1 is bounded_above iff b1 is summable ) )
proof end;

theorem Th21: :: SERIES_1:21
for b1 being Real_Sequence holds
( b1 is summable & ( for b2 being Nat holds 0 <= b1 . b2 ) implies 0 <= Sum b1 )
proof end;

theorem Th22: :: SERIES_1:22
for b1, b2 being Real_Sequence holds
( ( for b3 being Nat holds 0 <= b1 . b3 ) & b2 is summable & ex b3 being Nat st
for b4 being Nat holds
( b3 <= b4 implies b1 . b4 <= b2 . b4 ) implies b1 is summable )
proof end;

theorem Th23: :: SERIES_1:23
canceled;

theorem Th24: :: SERIES_1:24
for b1, b2 being Real_Sequence holds
( ( for b3 being Nat holds
( 0 <= b1 . b3 & b1 . b3 <= b2 . b3 ) ) & b2 is summable implies ( b1 is summable & Sum b1 <= Sum b2 ) )
proof end;

theorem Th25: :: SERIES_1:25
for b1 being Real_Sequence holds
( b1 is summable iff for b2 being real number holds
not ( 0 < b2 & ( for b3 being Nat holds
ex b4 being Nat st
( b3 <= b4 & not abs (((Partial_Sums b1) . b4) - ((Partial_Sums b1) . b3)) < b2 ) ) ) )
proof end;

theorem Th26: :: SERIES_1:26
for b1 being Nat
for b2 being real number holds
( b2 <> 1 implies (Partial_Sums (b2 GeoSeq )) . b1 = (1 - (b2 to_power (b1 + 1))) / (1 - b2) )
proof end;

theorem Th27: :: SERIES_1:27
for b1 being real number
for b2 being Real_Sequence holds
( b1 <> 1 & ( for b3 being Nat holds b2 . (b3 + 1) = b1 * (b2 . b3) ) implies for b3 being Nat holds (Partial_Sums b2) . b3 = ((b2 . 0) * (1 - (b1 to_power (b3 + 1)))) / (1 - b1) )
proof end;

theorem Th28: :: SERIES_1:28
for b1 being real number holds
( abs b1 < 1 implies ( b1 GeoSeq is summable & Sum (b1 GeoSeq ) = 1 / (1 - b1) ) )
proof end;

theorem Th29: :: SERIES_1:29
for b1 being real number
for b2 being Real_Sequence holds
( abs b1 < 1 & ( for b3 being Nat holds b2 . (b3 + 1) = b1 * (b2 . b3) ) implies ( b2 is summable & Sum b2 = (b2 . 0) / (1 - b1) ) )
proof end;

theorem Th30: :: SERIES_1:30
for b1, b2 being Real_Sequence holds
( ( for b3 being Nat holds
( b1 . b3 > 0 & b2 . b3 = (b1 . (b3 + 1)) / (b1 . b3) ) ) & b2 is convergent & lim b2 < 1 implies b1 is summable )
proof end;

theorem Th31: :: SERIES_1:31
for b1 being Real_Sequence holds
not ( ( for b2 being Nat holds
b1 . b2 > 0 ) & ex b2 being Nat st
for b3 being Nat holds
( b3 >= b2 implies (b1 . (b3 + 1)) / (b1 . b3) >= 1 ) & b1 is summable )
proof end;

theorem Th32: :: SERIES_1:32
for b1, b2 being Real_Sequence holds
( ( for b3 being Nat holds
( b1 . b3 >= 0 & b2 . b3 = b3 -root (b1 . b3) ) ) & b2 is convergent & lim b2 < 1 implies b1 is summable )
proof end;

theorem Th33: :: SERIES_1:33
for b1, b2 being Real_Sequence holds
not ( ( for b3 being Nat holds
( b1 . b3 >= 0 & b2 . b3 = b3 -root (b1 . b3) ) ) & ex b3 being Nat st
for b4 being Nat holds
( b3 <= b4 implies b2 . b4 >= 1 ) & b1 is summable )
proof end;

theorem Th34: :: SERIES_1:34
for b1, b2 being Real_Sequence holds
not ( ( for b3 being Nat holds
( b1 . b3 >= 0 & b2 . b3 = b3 -root (b1 . b3) ) ) & b2 is convergent & lim b2 > 1 & b1 is summable )
proof end;

definition
let c1, c2 be Nat;
redefine func to_power as c1 to_power c2 -> Nat;
coherence
c1 to_power c2 is Nat
proof end;
end;

theorem Th35: :: SERIES_1:35
for b1, b2 being Real_Sequence holds
( b1 is non-increasing & ( for b3 being Nat holds
( b1 . b3 >= 0 & b2 . b3 = (2 to_power b3) * (b1 . (2 to_power b3)) ) ) implies ( b1 is summable iff b2 is summable ) )
proof end;

theorem Th36: :: SERIES_1:36
for b1 being real number
for b2 being Real_Sequence holds
( b1 > 1 & ( for b3 being Nat holds
( b3 >= 1 implies b2 . b3 = 1 / (b3 to_power b1) ) ) implies b2 is summable )
proof end;

theorem Th37: :: SERIES_1:37
for b1 being real number
for b2 being Real_Sequence holds
not ( b1 <= 1 & ( for b3 being Nat holds
( b3 >= 1 implies b2 . b3 = 1 / (b3 to_power b1) ) ) & b2 is summable )
proof end;

definition
let c1 be Real_Sequence;
canceled;
attr a1 is absolutely_summable means :Def5: :: SERIES_1:def 5
abs a1 is summable;
end;

:: deftheorem Def4 SERIES_1:def 4 :
canceled;

:: deftheorem Def5 defines absolutely_summable SERIES_1:def 5 :
for b1 being Real_Sequence holds
( b1 is absolutely_summable iff abs b1 is summable );

theorem Th38: :: SERIES_1:38
canceled;

theorem Th39: :: SERIES_1:39
for b1 being Real_Sequence
for b2, b3 being Nat holds
( b2 <= b3 implies abs (((Partial_Sums b1) . b3) - ((Partial_Sums b1) . b2)) <= abs (((Partial_Sums (abs b1)) . b3) - ((Partial_Sums (abs b1)) . b2)) )
proof end;

theorem Th40: :: SERIES_1:40
for b1 being Real_Sequence holds
( b1 is absolutely_summable implies b1 is summable )
proof end;

theorem Th41: :: SERIES_1:41
for b1 being Real_Sequence holds
( ( for b2 being Nat holds 0 <= b1 . b2 ) & b1 is summable implies b1 is absolutely_summable )
proof end;

theorem Th42: :: SERIES_1:42
for b1, b2 being Real_Sequence holds
( ( for b3 being Nat holds
( b1 . b3 <> 0 & b2 . b3 = ((abs b1) . (b3 + 1)) / ((abs b1) . b3) ) ) & b2 is convergent & lim b2 < 1 implies b1 is absolutely_summable )
proof end;

theorem Th43: :: SERIES_1:43
for b1 being real number
for b2 being Real_Sequence holds
not ( b1 > 0 & ex b3 being Nat st
for b4 being Nat holds
( b4 >= b3 implies abs (b2 . b4) >= b1 ) & b2 is convergent & not lim b2 <> 0 )
proof end;

theorem Th44: :: SERIES_1:44
for b1 being Real_Sequence holds
not ( ( for b2 being Nat holds
b1 . b2 <> 0 ) & ex b2 being Nat st
for b3 being Nat holds
( b3 >= b2 implies ((abs b1) . (b3 + 1)) / ((abs b1) . b3) >= 1 ) & b1 is summable )
proof end;

theorem Th45: :: SERIES_1:45
for b1, b2 being Real_Sequence holds
( ( for b3 being Nat holds b1 . b3 = b3 -root ((abs b2) . b3) ) & b1 is convergent & lim b1 < 1 implies b2 is absolutely_summable )
proof end;

theorem Th46: :: SERIES_1:46
for b1, b2 being Real_Sequence holds
not ( ( for b3 being Nat holds b1 . b3 = b3 -root ((abs b2) . b3) ) & ex b3 being Nat st
for b4 being Nat holds
( b3 <= b4 implies b1 . b4 >= 1 ) & b2 is summable )
proof end;

theorem Th47: :: SERIES_1:47
for b1, b2 being Real_Sequence holds
not ( ( for b3 being Nat holds b1 . b3 = b3 -root ((abs b2) . b3) ) & b1 is convergent & lim b1 > 1 & b2 is summable )
proof end;

definition
let c1 be Real_Sequence;
let c2 be Nat;
func Sum c1,c2 -> Real equals :: SERIES_1:def 6
(Partial_Sums a1) . a2;
coherence
(Partial_Sums c1) . c2 is Real
;
end;

:: deftheorem Def6 defines Sum SERIES_1:def 6 :
for b1 being Real_Sequence
for b2 being Nat holds Sum b1,b2 = (Partial_Sums b1) . b2;

definition
let c1 be Real_Sequence;
let c2, c3 be Nat;
func Sum c1,c2,c3 -> Real equals :: SERIES_1:def 7
(Sum a1,a2) - (Sum a1,a3);
coherence
(Sum c1,c2) - (Sum c1,c3) is Real
;
end;

:: deftheorem Def7 defines Sum SERIES_1:def 7 :
for b1 being Real_Sequence
for b2, b3 being Nat holds Sum b1,b2,b3 = (Sum b1,b2) - (Sum b1,b3);