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
----------