I am looking to broaden my knowledge on infinite series I would like to know the proofs or the method with which mathematicians came to conclusion on why to use various tests and how did they propose and what arguments they used and also would like to more about several variable series convergence again not just follow a set of tests i want to know why and what logic they used to propose these
I would really appreciate a good book which discuss the concepts as deeply as possible
Theory and Application of Infinite Series by Konrad Knopp
I saw the book its goo but looking for a more modern book which includes the last 100 years work as well
Thanks
Jam
You should look at any intro real-analysis book. Series are fundamental to analysis, and I suspect that the best way to get sequences and series are from analysis texts.
You can look on MSE for analysis recommendations. I personally like Bartle's Elements of Real Analysis (for whatever reason, the showing price on amazon is nearly \$200, but if you look for more than 2 seconds you'll see you can get a new copy for \$25) or Rosenlicht's Introduction to Analysis (an extremely good yet inexpensive Dover book).
I don't know anything about Knopp's book.
But I'd also like to say that you might find it disappointing. All the tests for convergence either boil down to basic approximations and the ratio test (which comes from comparing to geometric series), some form of Dirichlet's test (a periodic divided by something that goes to infinity will converge - this underlies the alternating series test, for example), or something radically more complicated. And multivariate series are about the same, except poorly understood in general.
Finally, you probably won't find a book that both introduces you to the subject and contains recent development. The differences in level of sophistication are vast.