TITLE: What are some physical, geometric, or otherwise useful interpretations of divergent series?
QUESTION [9 upvotes]: I don't understand what ideas such as Abel, Cesàro summation or other types of sum 'regularization' help us describe. What is the practical application to discussing the 'sum' of sequences that are not convergent in the usual sense? 
What is the motivation to assigning a value to an otherwise divergent sequence, and further why is it a good idea to call whatever comes out a 'summation'?

REPLY [2 votes]: There is a lot of hype around about "values" of certain divergent series, like $\sum_{k=1}^\infty (-1)^kk$. But so far nobody has come up with a way of assigning a value to such a series which could be called "universal", or "canonical". 
It is, however, true that in some cases a divergent series $\sum_{k=1}^\infty a_k$ can be considered as a limiting case of a family of convergent series $$s(x):=\sum_{k=1}^\infty a_k(x)\qquad(x\in U)\ ,\tag{1}$$
whereby at the same time for all $k$ the limits
$$\lim_{x\to\xi} a_k(x)=a_k$$
exist, as well as the limit $\lim_{x\to\xi} s(x)=:\sigma$. One is then tempted to say that $\sum_{k=1}^\infty a_k=\sigma$, but making this conclusion is in fact voodoo mathematics.
Note that the "embedding" of the given series $\sum_{k=1}^\infty a_k$ into an environment $(1)$ involves many arbitrary choices by the master of ceremonies.