Reddit reviews Theory and Application of Infinite Series (Dover Books on Mathematics)
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You could look into Knopp's book on the subject. He has a section on divergent series. It's just a chapter as opposed to Hardy having an entire book on the topic, but it's more accessible.
Ahh, awesome. Though I would suggest Richard Courant's "Introduction to Calculus and Analysis I" due to my own bias, that book actually does not give the most complete exposition on infinite series. In fact, it is Richard Courant who suggested in a footnote that Konrad Knopp's "Theory and Application of Infinite Series" is the detailed treaties for infinite series, the book also touches on complex analysis (which is an amazing bonus, of course).
That is the book where I found an extremely straightforward derivation of the exponential generating function for the Bernoulli numbers, that's a very good book in my opinion (derivation on page 183).