Add new comment

Want facts and want them fast? Our Maths in a minute series explores key mathematical concepts in just a few words.
What do chocolate and mayonnaise have in common? It's maths! Find out how in this podcast featuring engineer Valerie Pinfield.
Is it possible to write unique music with the limited quantity of notes and chords available? We ask musician Oli Freke!
How can maths help to understand the Southern Ocean, a vital component of the Earth's climate system?
Was the mathematical modelling projecting the course of the pandemic too pessimistic, or were the projections justified? Matt Keeling tells our colleagues from SBIDER about the COVID models that fed into public policy.
PhD student Daniel Kreuter tells us about his work on the BloodCounts! project, which uses maths to make optimal use of the billions of blood tests performed every year around the globe.
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.