We all know there exists absolutely convergent series: it is one where it does not matter what the order of the series is upon summation of the infinite series.
I was wondering if there was some kind of similar property for when we assign values to divergent sums such that it does not matter which order we sum the terms in.
Questions
Do "absolutely divergent series" (series which whose value assigned does not matter on the ordering) exist? Is there any quick method to identify such divergent series?
Answer
You're missing the point of analytic continuation. Analytic continuation of a series doesn't have anything to do with the series i.e you're not changing the order of terms or manipulating the series in any way.
What you're actually doing is finding an analytic function which agrees with the given series where it converges but is also defined somewhere where the series diverges.
I've copied the stuff below the line from Wikipedia
If $f$ is an analytic function on a non-empty open subset $U$ of $\Bbb{C}$ then analytic continuations are unique in the following sense: if $V$ is the connected domain of two analytic functions $F_1$ and $F_2$ such that $U$ is contained in $V$ and for all $z$ in $U$
$$F_1(z)=F_2(z)=f(z)$$
then
$$F_1=F_2$$
on all of $V$.
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