Is there an algebra for divergent series summation operators? Let $D$ denote a divergent series and let $C$ denote a convergent series. 
Furthermore, let $s : $ { Series } $\to$ $\mathbb{C}$  be a regular, linear divergent series operator, which is either one of these operators: 
(the hyperlinks will direct you to the wiki page of the relevant summation method, not the person who invented/discovered it) 


*

* Borel  summation

*Abel summation

*Euler summation

*Cesàro summation

*Lambert summation

*Ramanujan summation

*Summing the series by means of Analytic continutation 

*Some  Regularization  method


I am wondering if there is any meaningful way to answer the following questions (Assuming $D_1 , D_2$ are summable with $s$):


*

*What does $s(D_1 + D_2)$ equal? Is it always equal to $s(D_2 + D_1)$ ? How does it relate to $s(D_1)$ and $s(D_2)$ ?

*What does $s(D_1 \cdot D_2) $ equal? Is it always equal to $s(D_2 \cdot D_1)$ ? How does it relate to $s(D_1)$ and $s(D_2)$ ?

*What happens when we add convergent series into the mix? And what if we're summing linear combinations of $n$ convergent and $m$ divergent series? 


Do the results differ for different summation methods, listed above? 
(This question was migrated from MSE. I also asked a somewhat similar question on MO once.)
 A: I'll try to offer as much knowledge as I can on this topic, which might not be that much. 
Firstly your linear form $s$ is defined on a strict linear subspace $L$ of the vector spaces of all formal series. Secondly you seem to mix formal numeric series and formal power series, which are distinct rings.
1) 2) By definition of the ring of formal (numeric or power) series over $\mathbb C$, the addition and product are commutative, so $s(D1+D2)=s(D2+D1)$ and $s(D1\cdot D2)=s(D2\cdot D1)$. Now if $s$ is $\mathbb C$-linear then $s(D1+D2)=s(D1)+s(D2)$. This is not true anymore for the product, even for convergent series as the product of two conditionnally convergent series might only be divergent. If both are absolutely convergent then the Cauchy-product formula shows that $s(D1\cdot D2)=s(D1)s(D2)$. This formula for convergent series is related to 3) below.
3) Every method you mention agrees with the usual sum for convergent series (except maybe Ramanujan's which I don't know about). So adding finitely many convergent series to the mix does not change anything by linearity. By the way the "summation by analytic continuation" is often refferred to as Mittag-Leffler's summation. Be careful though that it is not well defined due to the fact that most analytic continuations yield multivalued functions.
I think I should mention a nice paper by Lyubich which tries to axiomatically define what should be a coherent summation method for numeric series. The main property is that, following the afore-mentionned axioms, the sum $1+1+1+\cdots$ will never be assigned a finite value, i.e. it does not belong to $L$.
Also as far as I know the regularization process in QFT first transform a "series" of infinite quantities into a formal power series over $\mathbb C$ which may be (is) divergent, as explained somehow in this thread. In some cases the latter is a Borel-summable power series. See this survey (in French) by J.-P. Ramis for more details regarding this topic.
You might also want to learn more about Borel-summation for Gevrey power series through the works of Ramis, Sibuya, Balser and others and also through the mould/alien calculus developed by Écalle. An example is presented in my answer here.
