Q:

Prerequisites for understanding G.H. Hardy's 'Divergent Series'

I picked up a copy of G.H. Hardy's 'Divergent Series' a few days ago.  So far I love it, as I love the ideas associated with sequences and series, but I am finding it a bit difficult to understand.  I assumed I knew everything I would need to understand it(Calculus I/II, etc.), but perhaps not.  What would the possible prerequisites for understanding this book be?
Also, does anyone have any tactics for getting the best comprehension/understanding of the concepts in this book?
Thanks!

A:

Ah, yes, a book such as this is far away from undergrad curriculum, and even from basic grad-level math curriculum. It is not playing according to some quasi-orthodox set of rules, axioms, whatever one might say...
And, in my opinion, all the more charming and interesting because it is about "live" mathematics.
No, basic as-in-school ideas from calculus are wildly insufficient. Literally, "calculus" seems to teach us that "non-convergent " is meaningless. This is a gross over-statement and misrepresentation, ... as usual. Rather, a wiser interpretation of the situation is required, to see the genuine meaning that can be extracted (as opposed to just bailing out and declaring things "nonsense").
The naive form of "summing divergent series" seems to ask to capriciously assign values to "infinite sums" that don't have an "easy" (=convergent) sense. But, naturally, there are external constraints that prevent us from being completely capricious. However, in Hardy's time, there was more-limited vocabulary and framework for expressing "external constraints". Instead, by accident, his descriptive vocabulary would sound to us as "definitions"... specifically, lacking motivation or description of interaction/compatibility with "external requirements". 
In particular, he implicity, perhaps subliminally, and certainly rarely overtly in that text, there are implicit requirements of compatibility with meromorphic continuations (which Euler already understood at least subliminally), and other not-just-calculus issues.
So, again, no, basic calculus is not the context in which to understand that book. The standard curriculum (such as it is... sigh...) wouldn't provide the relevant formal (sigh...) background until second or third year of math-grad-school, if then. Small wonder it's hard to understand directly/naively.
Yet, yes, the book does have charms, and does not consciously try to oppress the reader...

