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);