A paragraph [Infinite Series] is added to the file:
https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download

.....
+-----------------+
| Infinite Series |
+-----------------+
Series::= S= Σ(n=0,k) a(n)= a(0)+ a(1)+ a(2) +... +a(k)
a(n) is called the general term, a(0),a(1),... the addend, summand or just
term. n is referred to as the index. Series S is the sum from the first term
a(0) to the last term a(k). The sum of those first terms (n<k) is called the
partial sum. "a(0)+...+a(k)" is called expanded form.

Infinite Series::= If the series S refers to infinite terms/addend (n=∞), S is
called an infinite series. Note that there are infinite(NEVER terminate)
addends. I.e. basically, the addition of addends cannot be completed in
finite steps by definition.

Operation Principle of Infinite Series: The last addend of the expanded form
(the index is ∞) must be shown to indicate the general term.

The arithmetic of the expanded form is the same as finite series:
Ex1: Let S= Σ(n=0,∞) a^n = 1+a+a^2+...+a^∞)
S= 1+a*(1+a+a^2+...+a^∞)- a*a^∞
<=> S= 1+a*S-a^(∞+1)
<=> S(1-a)=1-a^(∞+1)
<=> S= (1-a^(∞+1))/(1-a)

Ex2: Let S= Σ(n=1,∞) n = 1+2+3+...+n
S= 1+2+3+...+n // (1)
S= n+...+3+2+1 // (2)
2S= n*(n+1) // (1)+(2)
<=> S= n*(n+1)/2

If the last addend is missing, the expanded form is prone to magic tricks,
because the rearrangement of the expanded form may likely change the
definition of the series:
Ex1: S can be any number from a rearrangement:
S= Σ(n=1,∞) n= 1+2+3+... =1+1+1+1+...= (1+1)+(1+1+1)+...
= Σ(n=1,∞) n+1 // S is modified
(or S=(1+2)+(3+4)+... = Σ(n=1,∞) 4*n-1)

Ex2:
S=1+2+4+8+... // The last addend is omitted (ill-formed)
<=> S=1+2(1+2+4+8+...)
<=> S=1+2S
<=> S=-1

Last addend is shown:
S=1+2+4+8+...+2^∞
<=> S=1+2(1+2+4+...+2^(∞-1))
<=> S=1+2S-2^(∞+1)
<=> S=2^(∞+1)-1 // Lots of similar "magic calculation" deriving the result
// S=-1 can be found in youtube (from the omission of the
// term containing ∞).

Theorem1: s1=s2 <=> s1-s2=0

Theorem2: Σ(n=0,∞) a(n)= a(0)+ Σ(n=1,∞) a(n)
= a(∞)+ Σ(n=0,∞-1) a(n)

Theorem3: Σ(n=0,∞) f(n) ± Σ(n=0,∞) g(n) = Σ(n=0,∞) f(n)±g(n)
Theorem4: Σ(n=0,∞) c*f(n)= c*(Σ(n=0,∞) f(n))
Proof: Omitted (Can be derived from the expanded form. Trivial rules are also
omitted)

Basically, formula for finite series are also applicable to infinite series(
but mathematical inducion cannot prove such formula because by definition,
∞ means 'the procedure never terminate' and the Peano axiom is only valid in
finite steps).

Note: Many 'equations' of infinite series (esp. about π,e) can be proved
false by the theorems above. They are actually approximates (limits).
Ex: Σ(n=1,∞) 1/n² ≒ π²/6
Σ(n=0,∞) (-1^n)*(1/(2n+1)) ≒ π/4
Σ(n=0,∞) k^n/n! ≒ e^k
----------

On 04/24/2024 12:00 PM, wij wrote:
> A paragraph [Infinite Series] is added to the file:
> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>
> ....
> +-----------------+
> | Infinite Series |
> +-----------------+
> Series::= S= Σ(n=0,k) a(n)= a(0)+ a(1)+ a(2) +... +a(k)
> a(n) is called the general term, a(0),a(1),... the addend, summand or just
> term. n is referred to as the index. Series S is the sum from the first term
> a(0) to the last term a(k). The sum of those first terms (n<k) is called the
> partial sum. "a(0)+...+a(k)" is called expanded form.
>
> Infinite Series::= If the series S refers to infinite terms/addend (n=∞), S is
> called an infinite series. Note that there are infinite(NEVER terminate)
> addends. I.e. basically, the addition of addends cannot be completed in
> finite steps by definition.
>
> Operation Principle of Infinite Series: The last addend of the expanded form
> (the index is ∞) must be shown to indicate the general term.
>
> The arithmetic of the expanded form is the same as finite series:
> Ex1: Let S= Σ(n=0,∞) a^n = 1+a+a^2+...+a^∞)
> S= 1+a*(1+a+a^2+...+a^∞)- a*a^∞
> <=> S= 1+a*S-a^(∞+1)
> <=> S(1-a)=1-a^(∞+1)
> <=> S= (1-a^(∞+1))/(1-a)
>
> Ex2: Let S= Σ(n=1,∞) n = 1+2+3+...+n
> S= 1+2+3+...+n // (1)
> S= n+...+3+2+1 // (2)
> 2S= n*(n+1) // (1)+(2)
> <=> S= n*(n+1)/2
>
> If the last addend is missing, the expanded form is prone to magic tricks,
> because the rearrangement of the expanded form may likely change the
> definition of the series:
> Ex1: S can be any number from a rearrangement:
> S= Σ(n=1,∞) n= 1+2+3+... =1+1+1+1+...= (1+1)+(1+1+1)+...
> = Σ(n=1,∞) n+1 // S is modified
> (or S=(1+2)+(3+4)+... = Σ(n=1,∞) 4*n-1)
>
> Ex2:
> S=1+2+4+8+... // The last addend is omitted (ill-formed)
> <=> S=1+2(1+2+4+8+...)
> <=> S=1+2S
> <=> S=-1
>
> Last addend is shown:
> S=1+2+4+8+...+2^∞
> <=> S=1+2(1+2+4+...+2^(∞-1))
> <=> S=1+2S-2^(∞+1)
> <=> S=2^(∞+1)-1 // Lots of similar "magic calculation" deriving the result
> // S=-1 can be found in youtube (from the omission of the
> // term containing ∞).
>
> Theorem1: s1=s2 <=> s1-s2=0
>
> Theorem2: Σ(n=0,∞) a(n)= a(0)+ Σ(n=1,∞) a(n)
> = a(∞)+ Σ(n=0,∞-1) a(n)
>
> Theorem3: Σ(n=0,∞) f(n) ± Σ(n=0,∞) g(n) = Σ(n=0,∞) f(n)±g(n)
> Theorem4: Σ(n=0,∞) c*f(n)= c*(Σ(n=0,∞) f(n))
> Proof: Omitted (Can be derived from the expanded form. Trivial rules are also
> omitted)
>
> Basically, formula for finite series are also applicable to infinite series(
> but mathematical inducion cannot prove such formula because by definition,
> ∞ means 'the procedure never terminate' and the Peano axiom is only valid in
> finite steps).
>
> Note: Many 'equations' of infinite series (esp. about π,e) can be proved
> false by the theorems above. They are actually approximates (limits).
> Ex: Σ(n=1,∞) 1/n² ≒ π²/6
> Σ(n=0,∞) (-1^n)*(1/(2n+1)) ≒ π/4
> Σ(n=0,∞) k^n/n! ≒ e^k
> ----------
>
>

Observe the law(s) of large numbers.

On 4/24/2024 12:48 PM, Ross Finlayson wrote:
> On 04/24/2024 12:00 PM, wij wrote:
>>   A paragraph [Infinite Series] is added to the file:
>> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>>
>> ....
>> +-----------------+
>> | Infinite Series |
>> +-----------------+
>> Series::= S= Σ(n=0,k) a(n)= a(0)+ a(1)+ a(2) +... +a(k)
>>    a(n) is called the general term, a(0),a(1),... the addend, summand
>> or just
>>    term. n is referred to as the index. Series S is the sum from the
>> first term
>>    a(0) to the last term a(k). The sum of those first terms (n<k) is
>> called the
>>    partial sum. "a(0)+...+a(k)" is called expanded form.
>>
>> Infinite Series::= If the series S refers to infinite terms/addend
>> (n=∞), S is
>>    called an infinite series. Note that there are infinite(NEVER
>> terminate)
>>    addends. I.e. basically, the addition of addends cannot be
>> completed in
>>    finite steps by definition.
>>
>> Operation Principle of Infinite Series: The last addend of the
>> expanded form
>>    (the index is ∞) must be shown to indicate the general term.
>>
>>    The arithmetic of the expanded form is the same as finite series:
>>    Ex1: Let S= Σ(n=0,∞) a^n = 1+a+a^2+...+a^∞)
>>      S= 1+a*(1+a+a^2+...+a^∞)- a*a^∞
>>      <=> S= 1+a*S-a^(∞+1)
>>      <=> S(1-a)=1-a^(∞+1)
>>      <=> S= (1-a^(∞+1))/(1-a)
>>
>>    Ex2: Let S= Σ(n=1,∞) n = 1+2+3+...+n
>>      S= 1+2+3+...+n  // (1)
>>      S= n+...+3+2+1  // (2)
>>      2S= n*(n+1)     // (1)+(2)
>>      <=> S= n*(n+1)/2
>>
>>    If the last addend is missing, the expanded form is prone to magic
>> tricks,
>>    because the rearrangement of the expanded form may likely change the
>>    definition of the series:
>>    Ex1: S can be any number from a rearrangement:
>>         S= Σ(n=1,∞) n= 1+2+3+... =1+1+1+1+...= (1+1)+(1+1+1)+...
>>         = Σ(n=1,∞) n+1  // S is modified
>>           (or S=(1+2)+(3+4)+... = Σ(n=1,∞) 4*n-1)
>>
>>    Ex2:
>>      S=1+2+4+8+...     // The last addend is omitted (ill-formed)
>>      <=> S=1+2(1+2+4+8+...)
>>      <=> S=1+2S
>>      <=> S=-1
>>
>>    Last addend is shown:
>>      S=1+2+4+8+...+2^∞
>>      <=> S=1+2(1+2+4+...+2^(∞-1))
>>      <=> S=1+2S-2^(∞+1)
>>      <=> S=2^(∞+1)-1   // Lots of similar "magic calculation" deriving
>> the result
>>                        // S=-1 can be found in youtube (from the
>> omission of the
>>                        // term containing ∞).
>>
>> Theorem1: s1=s2 <=> s1-s2=0
>>
>> Theorem2: Σ(n=0,∞) a(n)= a(0)+ Σ(n=1,∞) a(n)
>>                      = a(∞)+ Σ(n=0,∞-1) a(n)
>>
>> Theorem3: Σ(n=0,∞) f(n) ± Σ(n=0,∞) g(n) = Σ(n=0,∞) f(n)±g(n)
>> Theorem4: Σ(n=0,∞) c*f(n)= c*(Σ(n=0,∞) f(n))
>>    Proof: Omitted (Can be derived from the expanded form. Trivial
>> rules are also
>>           omitted)
>>
>>    Basically, formula for finite series are also applicable to
>> infinite series(
>>    but mathematical inducion cannot prove such formula because by
>> definition,
>>    ∞ means 'the procedure never terminate' and the Peano axiom is only
>> valid in
>>    finite steps).
>>
>>    Note: Many 'equations' of infinite series (esp. about π,e) can be
>> proved
>>          false by the theorems above. They are actually approximates
>> (limits).
>>          Ex: Σ(n=1,∞) 1/n² ≒ π²/6
>>              Σ(n=0,∞) (-1^n)*(1/(2n+1)) ≒ π/4
>>              Σ(n=0,∞) k^n/n! ≒ e^k
>> ----------
>>
>>
>
> Observe the law(s) of large numbers.
>

Define a large number? What is large to you?

Am 24.04.2024 um 22:27 schrieb Chris M. Thomasson:
> On 4/24/2024 12:48 PM, Ross Finlayson wrote:
>> On 04/24/2024 12:00 PM, wij wrote:
>>>   A paragraph [Infinite Series] is added to the file:
>>> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>>>
>>> ....
>>> +-----------------+
>>> | Infinite Series |
>>> +-----------------+
>>> Series::= S= Σ(n=0,k) a(n)= a(0)+ a(1)+ a(2) +... +a(k)
>>>    a(n) is called the general term, a(0),a(1),... the addend, summand
>>> or just
>>>    term. n is referred to as the index. Series S is the sum from the
>>> first term
>>>    a(0) to the last term a(k). The sum of those first terms (n<k) is
>>> called the
>>>    partial sum. "a(0)+...+a(k)" is called expanded form.
>>>
>>> Infinite Series::= If the series S refers to infinite terms/addend
>>> (n=∞), S is
>>>    called an infinite series. Note that there are infinite(NEVER
>>> terminate)
>>>    addends. I.e. basically, the addition of addends cannot be
>>> completed in
>>>    finite steps by definition.
>>>
>>> Operation Principle of Infinite Series: The last addend of the
>>> expanded form
>>>    (the index is ∞) must be shown to indicate the general term.
>>>
>>>    The arithmetic of the expanded form is the same as finite series:
>>>    Ex1: Let S= Σ(n=0,∞) a^n = 1+a+a^2+...+a^∞)
>>>      S= 1+a*(1+a+a^2+...+a^∞)- a*a^∞
>>>      <=> S= 1+a*S-a^(∞+1)
>>>      <=> S(1-a)=1-a^(∞+1)
>>>      <=> S= (1-a^(∞+1))/(1-a)
>>>
>>>    Ex2: Let S= Σ(n=1,∞) n = 1+2+3+...+n
>>>      S= 1+2+3+...+n  // (1)
>>>      S= n+...+3+2+1  // (2)
>>>      2S= n*(n+1)     // (1)+(2)
>>>      <=> S= n*(n+1)/2
>>>
>>>    If the last addend is missing, the expanded form is prone to magic
>>> tricks,
>>>    because the rearrangement of the expanded form may likely change the
>>>    definition of the series:
>>>    Ex1: S can be any number from a rearrangement:
>>>         S= Σ(n=1,∞) n= 1+2+3+... =1+1+1+1+...= (1+1)+(1+1+1)+...
>>>         = Σ(n=1,∞) n+1  // S is modified
>>>           (or S=(1+2)+(3+4)+... = Σ(n=1,∞) 4*n-1)
>>>
>>>    Ex2:
>>>      S=1+2+4+8+...     // The last addend is omitted (ill-formed)
>>>      <=> S=1+2(1+2+4+8+...)
>>>      <=> S=1+2S
>>>      <=> S=-1
>>>
>>>    Last addend is shown:
>>>      S=1+2+4+8+...+2^∞
>>>      <=> S=1+2(1+2+4+...+2^(∞-1))
>>>      <=> S=1+2S-2^(∞+1)
>>>      <=> S=2^(∞+1)-1   // Lots of similar "magic calculation"
>>> deriving the result
>>>                        // S=-1 can be found in youtube (from the
>>> omission of the
>>>                        // term containing ∞).
>>>
>>> Theorem1: s1=s2 <=> s1-s2=0
>>>
>>> Theorem2: Σ(n=0,∞) a(n)= a(0)+ Σ(n=1,∞) a(n)
>>>                      = a(∞)+ Σ(n=0,∞-1) a(n)
>>>
>>> Theorem3: Σ(n=0,∞) f(n) ± Σ(n=0,∞) g(n) = Σ(n=0,∞) f(n)±g(n)
>>> Theorem4: Σ(n=0,∞) c*f(n)= c*(Σ(n=0,∞) f(n))
>>>    Proof: Omitted (Can be derived from the expanded form. Trivial
>>> rules are also
>>>           omitted)
>>>
>>>    Basically, formula for finite series are also applicable to
>>> infinite series(
>>>    but mathematical inducion cannot prove such formula because by
>>> definition,
>>>    ∞ means 'the procedure never terminate' and the Peano axiom is
>>> only valid in
>>>    finite steps).
>>>
>>>    Note: Many 'equations' of infinite series (esp. about π,e) can be
>>> proved
>>>          false by the theorems above. They are actually approximates
>>> (limits).
>>>          Ex: Σ(n=1,∞) 1/n² ≒ π²/6
>>>              Σ(n=0,∞) (-1^n)*(1/(2n+1)) ≒ π/4
>>>              Σ(n=0,∞) k^n/n! ≒ e^k
>>> ----------
>>>
>>>
>>
>> Observe the law(s) of large numbers.
>>
>
> Define a large number? What is large to you?

See: https://en.wikipedia.org/wiki/Law_of_large_numbers

(Though I doesn't make any sense in the present context.)

Am 24.04.2024 um 22:27 schrieb Chris M. Thomasson:
> On 4/24/2024 12:48 PM, Ross Finlayson wrote:
>> On 04/24/2024 12:00 PM, wij wrote:
>>>   A paragraph [Infinite Series] is added to the file:
>>> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>>>
>>> ....
>>> +-----------------+
>>> | Infinite Series |
>>> +-----------------+
>>> Series::= S= Σ(n=0,k) a(n)= a(0)+ a(1)+ a(2) +... +a(k)
>>>    a(n) is called the general term, a(0),a(1),... the addend, summand
>>> or just
>>>    term. n is referred to as the index. Series S is the sum from the
>>> first term
>>>    a(0) to the last term a(k). The sum of those first terms (n<k) is
>>> called the
>>>    partial sum. "a(0)+...+a(k)" is called expanded form.
>>>
>>> Infinite Series::= If the series S refers to infinite terms/addend
>>> (n=∞), S is
>>>    called an infinite series. Note that there are infinite(NEVER
>>> terminate)
>>>    addends. I.e. basically, the addition of addends cannot be
>>> completed in
>>>    finite steps by definition.
>>>
>>> Operation Principle of Infinite Series: The last addend of the
>>> expanded form
>>>    (the index is ∞) must be shown to indicate the general term.
>>>
>>>    The arithmetic of the expanded form is the same as finite series:
>>>    Ex1: Let S= Σ(n=0,∞) a^n = 1+a+a^2+...+a^∞)
>>>      S= 1+a*(1+a+a^2+...+a^∞)- a*a^∞
>>>      <=> S= 1+a*S-a^(∞+1)
>>>      <=> S(1-a)=1-a^(∞+1)
>>>      <=> S= (1-a^(∞+1))/(1-a)
>>>
>>>    Ex2: Let S= Σ(n=1,∞) n = 1+2+3+...+n
>>>      S= 1+2+3+...+n  // (1)
>>>      S= n+...+3+2+1  // (2)
>>>      2S= n*(n+1)     // (1)+(2)
>>>      <=> S= n*(n+1)/2
>>>
>>>    If the last addend is missing, the expanded form is prone to magic
>>> tricks,
>>>    because the rearrangement of the expanded form may likely change the
>>>    definition of the series:
>>>    Ex1: S can be any number from a rearrangement:
>>>         S= Σ(n=1,∞) n= 1+2+3+... =1+1+1+1+...= (1+1)+(1+1+1)+...
>>>         = Σ(n=1,∞) n+1  // S is modified
>>>           (or S=(1+2)+(3+4)+... = Σ(n=1,∞) 4*n-1)
>>>
>>>    Ex2:
>>>      S=1+2+4+8+...     // The last addend is omitted (ill-formed)
>>>      <=> S=1+2(1+2+4+8+...)
>>>      <=> S=1+2S
>>>      <=> S=-1
>>>
>>>    Last addend is shown:
>>>      S=1+2+4+8+...+2^∞
>>>      <=> S=1+2(1+2+4+...+2^(∞-1))
>>>      <=> S=1+2S-2^(∞+1)
>>>      <=> S=2^(∞+1)-1   // Lots of similar "magic calculation"
>>> deriving the result
>>>                        // S=-1 can be found in youtube (from the
>>> omission of the
>>>                        // term containing ∞).
>>>
>>> Theorem1: s1=s2 <=> s1-s2=0
>>>
>>> Theorem2: Σ(n=0,∞) a(n)= a(0)+ Σ(n=1,∞) a(n)
>>>                      = a(∞)+ Σ(n=0,∞-1) a(n)
>>>
>>> Theorem3: Σ(n=0,∞) f(n) ± Σ(n=0,∞) g(n) = Σ(n=0,∞) f(n)±g(n)
>>> Theorem4: Σ(n=0,∞) c*f(n)= c*(Σ(n=0,∞) f(n))
>>>    Proof: Omitted (Can be derived from the expanded form. Trivial
>>> rules are also
>>>           omitted)
>>>
>>>    Basically, formula for finite series are also applicable to
>>> infinite series(
>>>    but mathematical inducion cannot prove such formula because by
>>> definition,
>>>    ∞ means 'the procedure never terminate' and the Peano axiom is
>>> only valid in
>>>    finite steps).
>>>
>>>    Note: Many 'equations' of infinite series (esp. about π,e) can be
>>> proved
>>>          false by the theorems above. They are actually approximates
>>> (limits).
>>>          Ex: Σ(n=1,∞) 1/n² ≒ π²/6
>>>              Σ(n=0,∞) (-1^n)*(1/(2n+1)) ≒ π/4
>>>              Σ(n=0,∞) k^n/n! ≒ e^k
>>> ----------
>>>
>>>
>>
>> Observe the law(s) of large numbers.
>>
>
> Define a large number? What is large to you?

See: https://en.wikipedia.org/wiki/Law_of_large_numbers

(Though it doesn't make any sense in the present context.)

On 04/24/2024 01:27 PM, Chris M. Thomasson wrote:
> On 4/24/2024 12:48 PM, Ross Finlayson wrote:
>> On 04/24/2024 12:00 PM, wij wrote:
>>> A paragraph [Infinite Series] is added to the file:
>>> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>>>
>>>
>>> ....
>>> +-----------------+
>>> | Infinite Series |
>>> +-----------------+
>>> Series::= S= Σ(n=0,k) a(n)= a(0)+ a(1)+ a(2) +... +a(k)
>>> a(n) is called the general term, a(0),a(1),... the addend, summand
>>> or just
>>> term. n is referred to as the index. Series S is the sum from the
>>> first term
>>> a(0) to the last term a(k). The sum of those first terms (n<k) is
>>> called the
>>> partial sum. "a(0)+...+a(k)" is called expanded form.
>>>
>>> Infinite Series::= If the series S refers to infinite terms/addend
>>> (n=∞), S is
>>> called an infinite series. Note that there are infinite(NEVER
>>> terminate)
>>> addends. I.e. basically, the addition of addends cannot be
>>> completed in
>>> finite steps by definition.
>>>
>>> Operation Principle of Infinite Series: The last addend of the
>>> expanded form
>>> (the index is ∞) must be shown to indicate the general term.
>>>
>>> The arithmetic of the expanded form is the same as finite series:
>>> Ex1: Let S= Σ(n=0,∞) a^n = 1+a+a^2+...+a^∞)
>>> S= 1+a*(1+a+a^2+...+a^∞)- a*a^∞
>>> <=> S= 1+a*S-a^(∞+1)
>>> <=> S(1-a)=1-a^(∞+1)
>>> <=> S= (1-a^(∞+1))/(1-a)
>>>
>>> Ex2: Let S= Σ(n=1,∞) n = 1+2+3+...+n
>>> S= 1+2+3+...+n // (1)
>>> S= n+...+3+2+1 // (2)
>>> 2S= n*(n+1) // (1)+(2)
>>> <=> S= n*(n+1)/2
>>>
>>> If the last addend is missing, the expanded form is prone to magic
>>> tricks,
>>> because the rearrangement of the expanded form may likely change the
>>> definition of the series:
>>> Ex1: S can be any number from a rearrangement:
>>> S= Σ(n=1,∞) n= 1+2+3+... =1+1+1+1+...= (1+1)+(1+1+1)+...
>>> = Σ(n=1,∞) n+1 // S is modified
>>> (or S=(1+2)+(3+4)+... = Σ(n=1,∞) 4*n-1)
>>>
>>> Ex2:
>>> S=1+2+4+8+... // The last addend is omitted (ill-formed)
>>> <=> S=1+2(1+2+4+8+...)
>>> <=> S=1+2S
>>> <=> S=-1
>>>
>>> Last addend is shown:
>>> S=1+2+4+8+...+2^∞
>>> <=> S=1+2(1+2+4+...+2^(∞-1))
>>> <=> S=1+2S-2^(∞+1)
>>> <=> S=2^(∞+1)-1 // Lots of similar "magic calculation"
>>> deriving the result
>>> // S=-1 can be found in youtube (from the
>>> omission of the
>>> // term containing ∞).
>>>
>>> Theorem1: s1=s2 <=> s1-s2=0
>>>
>>> Theorem2: Σ(n=0,∞) a(n)= a(0)+ Σ(n=1,∞) a(n)
>>> = a(∞)+ Σ(n=0,∞-1) a(n)
>>>
>>> Theorem3: Σ(n=0,∞) f(n) ± Σ(n=0,∞) g(n) = Σ(n=0,∞) f(n)±g(n)
>>> Theorem4: Σ(n=0,∞) c*f(n)= c*(Σ(n=0,∞) f(n))
>>> Proof: Omitted (Can be derived from the expanded form. Trivial
>>> rules are also
>>> omitted)
>>>
>>> Basically, formula for finite series are also applicable to
>>> infinite series(
>>> but mathematical inducion cannot prove such formula because by
>>> definition,
>>> ∞ means 'the procedure never terminate' and the Peano axiom is
>>> only valid in
>>> finite steps).
>>>
>>> Note: Many 'equations' of infinite series (esp. about π,e) can be
>>> proved
>>> false by the theorems above. They are actually approximates
>>> (limits).
>>> Ex: Σ(n=1,∞) 1/n² ≒ π²/6
>>> Σ(n=0,∞) (-1^n)*(1/(2n+1)) ≒ π/4
>>> Σ(n=0,∞) k^n/n! ≒ e^k
>>> ----------
>>>
>>>
>>
>> Observe the law(s) of large numbers.
>>
>
> Define a large number? What is large to you?

Well, there's potential, practical, effective, and actual infinity.

Then, once upon a time, Zdislav Kovaric used to write to sci.math,
he details some "kinds" of infinity.

Then, there are various sorts law(s) of large numbers,
it's what the each of the laws are, for any or all large
numbers, it's the each of the law(s) of large numbers, for
the usual law of small numbers.

Usually, practically or effectively infinite are large enough.

Not always, ....

It sort of helps to have at least three definitions of continuous
domains, which together form a more replete and complete than the usual
definition of "complete".

Here these infinite expressions and their completions and their closure
of the forms, there was a pretty good discussion about series that
converge or diverge, and series that slowly converge or slowly diverge,
about a bunch of various kinds of criteria of convergence and
divergence, due various operations and their alternations,
in infinite expressions.

Four is a pretty good-sized number, ....

On 4/24/2024 2:04 PM, Moebius wrote:
> Am 24.04.2024 um 22:27 schrieb Chris M. Thomasson:
>> On 4/24/2024 12:48 PM, Ross Finlayson wrote:
>>> On 04/24/2024 12:00 PM, wij wrote:
>>>>   A paragraph [Infinite Series] is added to the file:
>>>> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>>>>
>>>> ....
>>>> +-----------------+
>>>> | Infinite Series |
>>>> +-----------------+
>>>> Series::= S= Σ(n=0,k) a(n)= a(0)+ a(1)+ a(2) +... +a(k)
>>>>    a(n) is called the general term, a(0),a(1),... the addend,
>>>> summand or just
>>>>    term. n is referred to as the index. Series S is the sum from the
>>>> first term
>>>>    a(0) to the last term a(k). The sum of those first terms (n<k) is
>>>> called the
>>>>    partial sum. "a(0)+...+a(k)" is called expanded form.
>>>>
>>>> Infinite Series::= If the series S refers to infinite terms/addend
>>>> (n=∞), S is
>>>>    called an infinite series. Note that there are infinite(NEVER
>>>> terminate)
>>>>    addends. I.e. basically, the addition of addends cannot be
>>>> completed in
>>>>    finite steps by definition.
>>>>
>>>> Operation Principle of Infinite Series: The last addend of the
>>>> expanded form
>>>>    (the index is ∞) must be shown to indicate the general term.
>>>>
>>>>    The arithmetic of the expanded form is the same as finite series:
>>>>    Ex1: Let S= Σ(n=0,∞) a^n = 1+a+a^2+...+a^∞)
>>>>      S= 1+a*(1+a+a^2+...+a^∞)- a*a^∞
>>>>      <=> S= 1+a*S-a^(∞+1)
>>>>      <=> S(1-a)=1-a^(∞+1)
>>>>      <=> S= (1-a^(∞+1))/(1-a)
>>>>
>>>>    Ex2: Let S= Σ(n=1,∞) n = 1+2+3+...+n
>>>>      S= 1+2+3+...+n  // (1)
>>>>      S= n+...+3+2+1  // (2)
>>>>      2S= n*(n+1)     // (1)+(2)
>>>>      <=> S= n*(n+1)/2
>>>>
>>>>    If the last addend is missing, the expanded form is prone to
>>>> magic tricks,
>>>>    because the rearrangement of the expanded form may likely change the
>>>>    definition of the series:
>>>>    Ex1: S can be any number from a rearrangement:
>>>>         S= Σ(n=1,∞) n= 1+2+3+... =1+1+1+1+...= (1+1)+(1+1+1)+...
>>>>         = Σ(n=1,∞) n+1  // S is modified
>>>>           (or S=(1+2)+(3+4)+... = Σ(n=1,∞) 4*n-1)
>>>>
>>>>    Ex2:
>>>>      S=1+2+4+8+...     // The last addend is omitted (ill-formed)
>>>>      <=> S=1+2(1+2+4+8+...)
>>>>      <=> S=1+2S
>>>>      <=> S=-1
>>>>
>>>>    Last addend is shown:
>>>>      S=1+2+4+8+...+2^∞
>>>>      <=> S=1+2(1+2+4+...+2^(∞-1))
>>>>      <=> S=1+2S-2^(∞+1)
>>>>      <=> S=2^(∞+1)-1   // Lots of similar "magic calculation"
>>>> deriving the result
>>>>                        // S=-1 can be found in youtube (from the
>>>> omission of the
>>>>                        // term containing ∞).
>>>>
>>>> Theorem1: s1=s2 <=> s1-s2=0
>>>>
>>>> Theorem2: Σ(n=0,∞) a(n)= a(0)+ Σ(n=1,∞) a(n)
>>>>                      = a(∞)+ Σ(n=0,∞-1) a(n)
>>>>
>>>> Theorem3: Σ(n=0,∞) f(n) ± Σ(n=0,∞) g(n) = Σ(n=0,∞) f(n)±g(n)
>>>> Theorem4: Σ(n=0,∞) c*f(n)= c*(Σ(n=0,∞) f(n))
>>>>    Proof: Omitted (Can be derived from the expanded form. Trivial
>>>> rules are also
>>>>           omitted)
>>>>
>>>>    Basically, formula for finite series are also applicable to
>>>> infinite series(
>>>>    but mathematical inducion cannot prove such formula because by
>>>> definition,
>>>>    ∞ means 'the procedure never terminate' and the Peano axiom is
>>>> only valid in
>>>>    finite steps).
>>>>
>>>>    Note: Many 'equations' of infinite series (esp. about π,e) can be
>>>> proved
>>>>          false by the theorems above. They are actually approximates
>>>> (limits).
>>>>          Ex: Σ(n=1,∞) 1/n² ≒ π²/6
>>>>              Σ(n=0,∞) (-1^n)*(1/(2n+1)) ≒ π/4
>>>>              Σ(n=0,∞) k^n/n! ≒ e^k
>>>> ----------
>>>>
>>>>
>>>
>>> Observe the law(s) of large numbers.
>>>
>>
>> Define a large number? What is large to you?
>
>
> See: https://en.wikipedia.org/wiki/Law_of_large_numbers
>
> (Though it doesn't make any sense in the present context.)
>

Can I say that the inverse of this very small number seems interesting
wrt a larger number? Keep in mind that it's unbounded:

https://youtu.be/0jGaio87u3A

In the abstract, beyond our finite self's, this does go on forever....
Fair enough?

On 4/24/2024 2:02 PM, Ross Finlayson wrote:
> On 04/24/2024 01:27 PM, Chris M. Thomasson wrote:
>> On 4/24/2024 12:48 PM, Ross Finlayson wrote:
>>> On 04/24/2024 12:00 PM, wij wrote:
>>>>   A paragraph [Infinite Series] is added to the file:
>>>> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>>>>
>>>>
>>>> ....
>>>> +-----------------+
>>>> | Infinite Series |
>>>> +-----------------+
>>>> Series::= S= Σ(n=0,k) a(n)= a(0)+ a(1)+ a(2) +... +a(k)
>>>>    a(n) is called the general term, a(0),a(1),... the addend, summand
>>>> or just
>>>>    term. n is referred to as the index. Series S is the sum from the
>>>> first term
>>>>    a(0) to the last term a(k). The sum of those first terms (n<k) is
>>>> called the
>>>>    partial sum. "a(0)+...+a(k)" is called expanded form.
>>>>
>>>> Infinite Series::= If the series S refers to infinite terms/addend
>>>> (n=∞), S is
>>>>    called an infinite series. Note that there are infinite(NEVER
>>>> terminate)
>>>>    addends. I.e. basically, the addition of addends cannot be
>>>> completed in
>>>>    finite steps by definition.
>>>>
>>>> Operation Principle of Infinite Series: The last addend of the
>>>> expanded form
>>>>    (the index is ∞) must be shown to indicate the general term.
>>>>
>>>>    The arithmetic of the expanded form is the same as finite series:
>>>>    Ex1: Let S= Σ(n=0,∞) a^n = 1+a+a^2+...+a^∞)
>>>>      S= 1+a*(1+a+a^2+...+a^∞)- a*a^∞
>>>>      <=> S= 1+a*S-a^(∞+1)
>>>>      <=> S(1-a)=1-a^(∞+1)
>>>>      <=> S= (1-a^(∞+1))/(1-a)
>>>>
>>>>    Ex2: Let S= Σ(n=1,∞) n = 1+2+3+...+n
>>>>      S= 1+2+3+...+n  // (1)
>>>>      S= n+...+3+2+1  // (2)
>>>>      2S= n*(n+1)     // (1)+(2)
>>>>      <=> S= n*(n+1)/2
>>>>
>>>>    If the last addend is missing, the expanded form is prone to magic
>>>> tricks,
>>>>    because the rearrangement of the expanded form may likely change the
>>>>    definition of the series:
>>>>    Ex1: S can be any number from a rearrangement:
>>>>         S= Σ(n=1,∞) n= 1+2+3+... =1+1+1+1+...= (1+1)+(1+1+1)+...
>>>>         = Σ(n=1,∞) n+1  // S is modified
>>>>           (or S=(1+2)+(3+4)+... = Σ(n=1,∞) 4*n-1)
>>>>
>>>>    Ex2:
>>>>      S=1+2+4+8+...     // The last addend is omitted (ill-formed)
>>>>      <=> S=1+2(1+2+4+8+...)
>>>>      <=> S=1+2S
>>>>      <=> S=-1
>>>>
>>>>    Last addend is shown:
>>>>      S=1+2+4+8+...+2^∞
>>>>      <=> S=1+2(1+2+4+...+2^(∞-1))
>>>>      <=> S=1+2S-2^(∞+1)
>>>>      <=> S=2^(∞+1)-1   // Lots of similar "magic calculation"
>>>> deriving the result
>>>>                        // S=-1 can be found in youtube (from the
>>>> omission of the
>>>>                        // term containing ∞).
>>>>
>>>> Theorem1: s1=s2 <=> s1-s2=0
>>>>
>>>> Theorem2: Σ(n=0,∞) a(n)= a(0)+ Σ(n=1,∞) a(n)
>>>>                      = a(∞)+ Σ(n=0,∞-1) a(n)
>>>>
>>>> Theorem3: Σ(n=0,∞) f(n) ± Σ(n=0,∞) g(n) = Σ(n=0,∞) f(n)±g(n)
>>>> Theorem4: Σ(n=0,∞) c*f(n)= c*(Σ(n=0,∞) f(n))
>>>>    Proof: Omitted (Can be derived from the expanded form. Trivial
>>>> rules are also
>>>>           omitted)
>>>>
>>>>    Basically, formula for finite series are also applicable to
>>>> infinite series(
>>>>    but mathematical inducion cannot prove such formula because by
>>>> definition,
>>>>    ∞ means 'the procedure never terminate' and the Peano axiom is
>>>> only valid in
>>>>    finite steps).
>>>>
>>>>    Note: Many 'equations' of infinite series (esp. about π,e) can be
>>>> proved
>>>>          false by the theorems above. They are actually approximates
>>>> (limits).
>>>>          Ex: Σ(n=1,∞) 1/n² ≒ π²/6
>>>>              Σ(n=0,∞) (-1^n)*(1/(2n+1)) ≒ π/4
>>>>              Σ(n=0,∞) k^n/n! ≒ e^k
>>>> ----------
>>>>
>>>>
>>>
>>> Observe the law(s) of large numbers.
>>>
>>
>> Define a large number? What is large to you?
>
> Well, there's potential, practical, effective, and actual infinity.
>
> Then, once upon a time, Zdislav Kovaric used to write to sci.math,
> he details some "kinds" of infinity.
>
> Then, there are various sorts law(s) of large numbers,
> it's what the each of the laws are, for any or all large
> numbers, it's the each of the law(s) of large numbers, for
> the usual law of small numbers.
>
> Usually, practically or effectively infinite are large enough.
>
> Not always, ....
>
> It sort of helps to have at least three definitions of continuous
> domains, which together form a more replete and complete than the usual
> definition of "complete".
>
> Here these infinite expressions and their completions and their closure
> of the forms, there was a pretty good discussion about series that
> converge or diverge, and series that slowly converge or slowly diverge,
> about a bunch of various kinds of criteria of convergence and
> divergence, due various operations and their alternations,
> in infinite expressions.
>
>
> Four is a pretty good-sized number, ....
>

Only a little zoom on a Newton Set:

https://youtu.be/9V0Bj3K4gXs

On 4/24/2024 12:00 PM, wij wrote:
[...]

https://youtu.be/bpBvK-VhSjA

Am 24.04.2024 um 23:22 schrieb Chris M. Thomasson:
> On 4/24/2024 2:04 PM, Moebius wrote:
>> Am 24.04.2024 um 22:27 schrieb Chris M. Thomasson:
>>> On 4/24/2024 12:48 PM, Ross Finlayson wrote:
>>>> On 04/24/2024 12:00 PM, wij wrote:
>>>>>   A paragraph [Infinite Series] is added to the file:
>>>>> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>>>>>
>>>>> ....
>>>>> +-----------------+
>>>>> | Infinite Series |
>>>>> +-----------------+
>>>>> Series::= S= Σ(n=0,k) a(n)= a(0)+ a(1)+ a(2) +... +a(k)
>>>>>    a(n) is called the general term, a(0),a(1),... the addend,
>>>>> summand or just
>>>>>    term. n is referred to as the index. Series S is the sum from
>>>>> the first term
>>>>>    a(0) to the last term a(k). The sum of those first terms (n<k)
>>>>> is called the
>>>>>    partial sum. "a(0)+...+a(k)" is called expanded form.
>>>>>
>>>>> Infinite Series::= If the series S refers to infinite terms/addend
>>>>> (n=∞), S is
>>>>>    called an infinite series. Note that there are infinite(NEVER
>>>>> terminate)
>>>>>    addends. I.e. basically, the addition of addends cannot be
>>>>> completed in
>>>>>    finite steps by definition.
>>>>>
>>>>> Operation Principle of Infinite Series: The last addend of the
>>>>> expanded form
>>>>>    (the index is ∞) must be shown to indicate the general term.
>>>>>
>>>>>    The arithmetic of the expanded form is the same as finite series:
>>>>>    Ex1: Let S= Σ(n=0,∞) a^n = 1+a+a^2+...+a^∞)
>>>>>      S= 1+a*(1+a+a^2+...+a^∞)- a*a^∞
>>>>>      <=> S= 1+a*S-a^(∞+1)
>>>>>      <=> S(1-a)=1-a^(∞+1)
>>>>>      <=> S= (1-a^(∞+1))/(1-a)
>>>>>
>>>>>    Ex2: Let S= Σ(n=1,∞) n = 1+2+3+...+n
>>>>>      S= 1+2+3+...+n  // (1)
>>>>>      S= n+...+3+2+1  // (2)
>>>>>      2S= n*(n+1)     // (1)+(2)
>>>>>      <=> S= n*(n+1)/2
>>>>>
>>>>>    If the last addend is missing, the expanded form is prone to
>>>>> magic tricks,
>>>>>    because the rearrangement of the expanded form may likely change
>>>>> the
>>>>>    definition of the series:
>>>>>    Ex1: S can be any number from a rearrangement:
>>>>>         S= Σ(n=1,∞) n= 1+2+3+... =1+1+1+1+...= (1+1)+(1+1+1)+...
>>>>>         = Σ(n=1,∞) n+1  // S is modified
>>>>>           (or S=(1+2)+(3+4)+... = Σ(n=1,∞) 4*n-1)
>>>>>
>>>>>    Ex2:
>>>>>      S=1+2+4+8+...     // The last addend is omitted (ill-formed)
>>>>>      <=> S=1+2(1+2+4+8+...)
>>>>>      <=> S=1+2S
>>>>>      <=> S=-1
>>>>>
>>>>>    Last addend is shown:
>>>>>      S=1+2+4+8+...+2^∞
>>>>>      <=> S=1+2(1+2+4+...+2^(∞-1))
>>>>>      <=> S=1+2S-2^(∞+1)
>>>>>      <=> S=2^(∞+1)-1   // Lots of similar "magic calculation"
>>>>> deriving the result
>>>>>                        // S=-1 can be found in youtube (from the
>>>>> omission of the
>>>>>                        // term containing ∞).
>>>>>
>>>>> Theorem1: s1=s2 <=> s1-s2=0
>>>>>
>>>>> Theorem2: Σ(n=0,∞) a(n)= a(0)+ Σ(n=1,∞) a(n)
>>>>>                      = a(∞)+ Σ(n=0,∞-1) a(n)
>>>>>
>>>>> Theorem3: Σ(n=0,∞) f(n) ± Σ(n=0,∞) g(n) = Σ(n=0,∞) f(n)±g(n)
>>>>> Theorem4: Σ(n=0,∞) c*f(n)= c*(Σ(n=0,∞) f(n))
>>>>>    Proof: Omitted (Can be derived from the expanded form. Trivial
>>>>> rules are also
>>>>>           omitted)
>>>>>
>>>>>    Basically, formula for finite series are also applicable to
>>>>> infinite series(
>>>>>    but mathematical inducion cannot prove such formula because by
>>>>> definition,
>>>>>    ∞ means 'the procedure never terminate' and the Peano axiom is
>>>>> only valid in
>>>>>    finite steps).
>>>>>
>>>>>    Note: Many 'equations' of infinite series (esp. about π,e) can
>>>>> be proved
>>>>>          false by the theorems above. They are actually
>>>>> approximates (limits).
>>>>>          Ex: Σ(n=1,∞) 1/n² ≒ π²/6
>>>>>              Σ(n=0,∞) (-1^n)*(1/(2n+1)) ≒ π/4
>>>>>              Σ(n=0,∞) k^n/n! ≒ e^k
>>>>> ----------
>>>>>
>>>>>
>>>>
>>>> Observe the law(s) of large numbers.
>>>>
>>>
>>> Define a large number? What is large to you?
>>
>>
>> See: https://en.wikipedia.org/wiki/Law_of_large_numbers
>>
>> (Though it doesn't make any sense in the present context.)
>>
>
> Can I say that the inverse of this very small number seems interesting
> wrt a larger number? Keep in mind that it's unbounded:
>
> https://youtu.be/0jGaio87u3A

Holy shit, I swear, I only took a tiny piece!!!

> In the abstract, beyond our finite self's, this does go on forever....
> Fair enough?

Sure - at least in the "mathematical reality". :-P

wij was thinking very hard :
> A paragraph [Infinite Series] is added to the file:
> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>
> ....
> +-----------------+
>> Infinite Series |
> +-----------------+
> Series::= S= Σ(n=0,k) a(n)= a(0)+ a(1)+ a(2) +... +a(k)
> a(n) is called the general term, a(0),a(1),... the addend, summand or just
> term. n is referred to as the index. Series S is the sum from the first
> term a(0) to the last term a(k). The sum of those first terms (n<k) is
> called the partial sum. "a(0)+...+a(k)" is called expanded form.
>
> Infinite Series::= If the series S refers to infinite terms/addend (n=∞), S
> is called an infinite series. Note that there are infinite(NEVER terminate)
> addends. I.e. basically, the addition of addends cannot be completed in
> finite steps by definition.
>
> Operation Principle of Infinite Series: The last addend of the expanded form
> (the index is ∞) must be shown to indicate the general term.
>
> The arithmetic of the expanded form is the same as finite series:
> Ex1: Let S= Σ(n=0,∞) a^n = 1+a+a^2+...+a^∞)
> S= 1+a*(1+a+a^2+...+a^∞)- a*a^∞
> <=> S= 1+a*S-a^(∞+1)
> <=> S(1-a)=1-a^(∞+1)
> <=> S= (1-a^(∞+1))/(1-a)
>
> Ex2: Let S= Σ(n=1,∞) n = 1+2+3+...+n
> S= 1+2+3+...+n // (1)
> S= n+...+3+2+1 // (2)
> 2S= n*(n+1) // (1)+(2)
> <=> S= n*(n+1)/2
>
> If the last addend is missing, the expanded form is prone to magic tricks,
> because the rearrangement of the expanded form may likely change the
> definition of the series:
> Ex1: S can be any number from a rearrangement:
> S= Σ(n=1,∞) n= 1+2+3+... =1+1+1+1+...= (1+1)+(1+1+1)+...
> = Σ(n=1,∞) n+1 // S is modified
> (or S=(1+2)+(3+4)+... = Σ(n=1,∞) 4*n-1)
>
> Ex2:
> S=1+2+4+8+... // The last addend is omitted (ill-formed)
> <=> S=1+2(1+2+4+8+...)
> <=> S=1+2S
> <=> S=-1
>
> Last addend is shown:
> S=1+2+4+8+...+2^∞
> <=> S=1+2(1+2+4+...+2^(∞-1))
> <=> S=1+2S-2^(∞+1)
> <=> S=2^(∞+1)-1 // Lots of similar "magic calculation" deriving the
> result // S=-1 can be found in youtube (from the
> omission of the // term containing ∞).
>
> Theorem1: s1=s2 <=> s1-s2=0
>
> Theorem2: Σ(n=0,∞) a(n)= a(0)+ Σ(n=1,∞) a(n)
> = a(∞)+ Σ(n=0,∞-1) a(n)
>
> Theorem3: Σ(n=0,∞) f(n) ± Σ(n=0,∞) g(n) = Σ(n=0,∞) f(n)±g(n)
> Theorem4: Σ(n=0,∞) c*f(n)= c*(Σ(n=0,∞) f(n))
> Proof: Omitted (Can be derived from the expanded form. Trivial rules are
> also omitted)
>
> Basically, formula for finite series are also applicable to infinite
> series( but mathematical inducion cannot prove such formula because by
> definition, ∞ means 'the procedure never terminate' and the Peano axiom is
> only valid in finite steps).
>
> Note: Many 'equations' of infinite series (esp. about π,e) can be proved
> false by the theorems above. They are actually approximates (limits).
> Ex: Σ(n=1,∞) 1/n² ≒ π²/6
> Σ(n=0,∞) (-1^n)*(1/(2n+1)) ≒ π/4
> Σ(n=0,∞) k^n/n! ≒ e^k
> ----------

For a finite sequence, add the finitely many terms by simple addition.
If the sequence is infinite, you need a series to obtain a sum. The
word series already implies an attempt at summing infinitely many
terms.

On Wed, 2024-04-24 at 19:40 -0400, FromTheRafters wrote:
> wij was thinking very hard :
> >  A paragraph [Infinite Series] is added to the file:
> > https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
> >
> > ....
> > +-----------------+
> > > Infinite Series |
> > +-----------------+
> > Series::= S= Σ(n=0,k) a(n)= a(0)+ a(1)+ a(2) +... +a(k)
> >   a(n) is called the general term, a(0),a(1),... the addend, summand or just
> >   term. n is referred to as the index. Series S is the sum from the first
> > term   a(0) to the last term a(k). The sum of those first terms (n<k) is
> > called the   partial sum. "a(0)+...+a(k)" is called expanded form.
> >
> > Infinite Series::= If the series S refers to infinite terms/addend (n=∞), S
> > is   called an infinite series. Note that there are infinite(NEVER terminate)
> >   addends. I.e. basically, the addition of addends cannot be completed in
> >   finite steps by definition.
> >
> > Operation Principle of Infinite Series: The last addend of the expanded form
> >   (the index is ∞) must be shown to indicate the general term.
> >
> >   The arithmetic of the expanded form is the same as finite series:
> >   Ex1: Let S= Σ(n=0,∞) a^n = 1+a+a^2+...+a^∞)
> >     S= 1+a*(1+a+a^2+...+a^∞)- a*a^∞
> >     <=> S= 1+a*S-a^(∞+1)
> >     <=> S(1-a)=1-a^(∞+1)
> >     <=> S= (1-a^(∞+1))/(1-a)
> >
> >   Ex2: Let S= Σ(n=1,∞) n = 1+2+3+...+n
> >     S= 1+2+3+...+n  // (1)
> >     S= n+...+3+2+1  // (2)
> >     2S= n*(n+1)     // (1)+(2)
> >     <=> S= n*(n+1)/2
> >
> >   If the last addend is missing, the expanded form is prone to magic tricks,
> >   because the rearrangement of the expanded form may likely change the
> >   definition of the series:
> >   Ex1: S can be any number from a rearrangement:
> >        S= Σ(n=1,∞) n= 1+2+3+... =1+1+1+1+...= (1+1)+(1+1+1)+...
> >        = Σ(n=1,∞) n+1  // S is modified
> >          (or S=(1+2)+(3+4)+.... = Σ(n=1,∞) 4*n-1)
> >
> >   Ex2:
> >     S=1+2+4+8+...     // The last addend is omitted (ill-formed)
> >     <=> S=1+2(1+2+4+8+...)
> >     <=> S=1+2S
> >     <=> S=-1
> >
> >   Last addend is shown:
> >     S=1+2+4+8+...+2^∞
> >     <=> S=1+2(1+2+4+...+2^(∞-1))
> >     <=> S=1+2S-2^(∞+1)
> >     <=> S=2^(∞+1)-1   // Lots of similar "magic calculation" deriving the
> > result                       // S=-1 can be found in youtube (from the
> > omission of the                       // term containing ∞).
> >
> > Theorem1: s1=s2 <=> s1-s2=0
> >
> > Theorem2: Σ(n=0,∞) a(n)= a(0)+ Σ(n=1,∞) a(n)
> >                     = a(∞)+ Σ(n=0,∞-1) a(n)
> >
> > Theorem3: Σ(n=0,∞) f(n) ± Σ(n=0,∞) g(n) = Σ(n=0,∞) f(n)±g(n)
> > Theorem4: Σ(n=0,∞) c*f(n)= c*(Σ(n=0,∞) f(n))
> >   Proof: Omitted (Can be derived from the expanded form. Trivial rules are
> > also          omitted)
> >
> >   Basically, formula for finite series are also applicable to infinite
> > series(   but mathematical inducion cannot prove such formula because by
> > definition,   ∞ means 'the procedure never terminate' and the Peano axiom is
> > only valid in   finite steps).
> >
> >   Note: Many 'equations' of infinite series (esp. about π,e) can be proved
> >         false by the theorems above. They are actually approximates (limits).
> >         Ex: Σ(n=1,∞) 1/n² ≒ π²/6
> >             Σ(n=0,∞) (-1^n)*(1/(2n+1)) ≒ π/4
> >             Σ(n=0,∞) k^n/n! ≒ e^k
> > ----------
>
> For a finite sequence, add the finitely many terms by simple addition.
> If the sequence is infinite, you need a series to obtain a sum. The
> word series already implies an attempt at summing infinitely many
> terms.

Not sure what you were talking about.

wij was thinking very hard :
> On Wed, 2024-04-24 at 19:40 -0400, FromTheRafters wrote:
>> wij was thinking very hard :
>>>  A paragraph [Infinite Series] is added to the file:
>>> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>>>
>>> ....
>>> +-----------------+
>>>> Infinite Series |
>>> +-----------------+
>>> Series::= S= Σ(n=0,k) a(n)= a(0)+ a(1)+ a(2) +... +a(k)
>>>   a(n) is called the general term, a(0),a(1),... the addend, summand or
>>> just   term. n is referred to as the index. Series S is the sum from the
>>> first term   a(0) to the last term a(k). The sum of those first terms
>>> (n<k) is called the   partial sum. "a(0)+...+a(k)" is called expanded
>>> form.
>>>
>>> Infinite Series::= If the series S refers to infinite terms/addend (n=∞), S
>>> is   called an infinite series. Note that there are infinite(NEVER
>>> terminate)   addends. I.e. basically, the addition of addends cannot be
>>> completed in   finite steps by definition.
>>>
>>> Operation Principle of Infinite Series: The last addend of the expanded
>>> form   (the index is ∞) must be shown to indicate the general term.
>>>
>>>   The arithmetic of the expanded form is the same as finite series:
>>>   Ex1: Let S= Σ(n=0,∞) a^n = 1+a+a^2+...+a^∞)
>>>     S= 1+a*(1+a+a^2+...+a^∞)- a*a^∞
>>>     <=> S= 1+a*S-a^(∞+1)
>>>     <=> S(1-a)=1-a^(∞+1)
>>>     <=> S= (1-a^(∞+1))/(1-a)
>>>
>>>   Ex2: Let S= Σ(n=1,∞) n = 1+2+3+...+n
>>>     S= 1+2+3+...+n  // (1)
>>>     S= n+...+3+2+1  // (2)
>>>     2S= n*(n+1)     // (1)+(2)
>>>     <=> S= n*(n+1)/2
>>>
>>>   If the last addend is missing, the expanded form is prone to magic
>>> tricks,   because the rearrangement of the expanded form may likely change
>>> the   definition of the series:
>>>   Ex1: S can be any number from a rearrangement:
>>>        S= Σ(n=1,∞) n= 1+2+3+... =1+1+1+1+...= (1+1)+(1+1+1)+...
>>>        = Σ(n=1,∞) n+1  // S is modified
>>>          (or S=(1+2)+(3+4)+... = Σ(n=1,∞) 4*n-1)
>>>
>>>   Ex2:
>>>     S=1+2+4+8+...     // The last addend is omitted (ill-formed)
>>>     <=> S=1+2(1+2+4+8+...)
>>>     <=> S=1+2S
>>>     <=> S=-1
>>>
>>>   Last addend is shown:
>>>     S=1+2+4+8+...+2^∞
>>>     <=> S=1+2(1+2+4+...+2^(∞-1))
>>>     <=> S=1+2S-2^(∞+1)
>>>     <=> S=2^(∞+1)-1   // Lots of similar "magic calculation" deriving the
>>> result                       // S=-1 can be found in youtube (from the
>>> omission of the                       // term containing ∞).
>>>
>>> Theorem1: s1=s2 <=> s1-s2=0
>>>
>>> Theorem2: Σ(n=0,∞) a(n)= a(0)+ Σ(n=1,∞) a(n)
>>>                     = a(∞)+ Σ(n=0,∞-1) a(n)
>>>
>>> Theorem3: Σ(n=0,∞) f(n) ± Σ(n=0,∞) g(n) = Σ(n=0,∞) f(n)±g(n)
>>> Theorem4: Σ(n=0,∞) c*f(n)= c*(Σ(n=0,∞) f(n))
>>>   Proof: Omitted (Can be derived from the expanded form. Trivial rules are
>>> also          omitted)
>>>
>>>   Basically, formula for finite series are also applicable to infinite
>>> series(   but mathematical inducion cannot prove such formula because by
>>> definition,   ∞ means 'the procedure never terminate' and the Peano axiom
>>> is only valid in   finite steps).
>>>
>>>   Note: Many 'equations' of infinite series (esp. about π,e) can be proved
>>>         false by the theorems above. They are actually approximates
>>> (limits).         Ex: Σ(n=1,∞) 1/n² ≒ π²/6
>>>             Σ(n=0,∞) (-1^n)*(1/(2n+1)) ≒ π/4
>>>             Σ(n=0,∞) k^n/n! ≒ e^k
>>> ----------
>>
>> For a finite sequence, add the finitely many terms by simple addition.
>> If the sequence is infinite, you need a series to obtain a sum. The
>> word series already implies an attempt at summing infinitely many
>> terms.
>
> Not sure what you were talking about.

https://proofwiki.org/wiki/Definition:Series/Also_known_as