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?
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?
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?
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