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.)

conditionsfor divergent series (as opposed to the ones listed on wikipedia), or do you think I ought to ask different... (cont'd) – Max Muller Apr 26 '13 at 20:38questionsabout them with the conditions as they are now? – Max Muller Apr 26 '13 at 20:38