When you say that the SIGMA of a sequence of terms 'is' V? You are ASSIGNING a value V to a sequence.
When you say "this sequence's sum is V" : this is a misuse of language!
Let's us call it an ASSIGNEMENT theory:
Each theory is suppose to obey AXIOMS ( dixit Hardy) , SUM is a function from a space of sequences on R to some (semi)ring R (i.e. N,Z,Q,R,C)
For any sequence s,s'
A1) SUM ( k.s ) = k.SUM(s) for any scalar k in R
A1) SUM ( s + s' ) = SUM(s) + SUM(s')
A3) SUM ( s shifted once) = SUM (s) - first term of s
In plain language the axioms guarantee the compatibility of SUM with adding , scalar multiplication and finite term shifting.

Examples of theory
T1) The zero theory : Any infinite sequence sums to ZERO. Obeys axioms 1,2,3
T2) The infinite theory: Any infinite sequence sums to zero if all terms are zero , else to +infinity (resp -infinity) when first non zero term is positive (respectively negative).
Obeys axioms A1 only.
T3) Classical theory : sums to V if partial sums tend to V ...Obeys axioms 1,2,3
T4) Cesaro sums ... Obeys axioms 1,2,3
T5) Ghandi sums any one that fit Axioms 1,2,3
Remark R1 : using R = N the postive integer assigning +infinity to any serie will obeys AXIOM 1,2,3

In conclusion:
Remark R2 : The contradictions shown in sum posts do not respect all axioms in their arguments.
A serie is ASSIGNED a value NOT EQUAL to that value.

When you say that the SIGMA of a sequence of terms 'is' V? You are

ASSIGNINGa value V to a sequence.When you say "this sequence's sum is V" : this is a misuse of language!

Let's us call it an ASSIGNEMENT theory:

Each theory is suppose to obey AXIOMS ( dixit Hardy) , SUM is a function from a space of sequences on R to some (semi)ring R (i.e. N,Z,Q,R,C)

For any sequence s,s'

A1) SUM ( k.s ) = k.SUM(s) for any scalar k in R

A1) SUM ( s + s' ) = SUM(s) + SUM(s')

A3) SUM ( s shifted once) = SUM (s) - first term of s

In plain language the axioms guarantee the compatibility of SUM with adding , scalar multiplication and finite term shifting.

Examples of theory

T1) The zero theory : Any infinite sequence sums to ZERO. Obeys axioms 1,2,3

T2) The infinite theory: Any infinite sequence sums to zero if all terms are zero , else to +infinity (resp -infinity) when first non zero term is positive (respectively negative).

Obeys axioms A1 only.

T3) Classical theory : sums to V if partial sums tend to V ...Obeys axioms 1,2,3

T4) Cesaro sums ... Obeys axioms 1,2,3

T5) Ghandi sums any one that fit Axioms 1,2,3

Remark R1 : using R = N the postive integer assigning +infinity to any serie will obeys AXIOM 1,2,3

In conclusion:Remark R2 : The contradictions shown in sum posts do not respect all axioms in their arguments.

A serie is

ASSIGNEDa valueNOT EQUALto that value.