First of all I'm interested in methods of convergence (for sums and integrals) so are there good books which tackle these methods one after the other and with examples that are not too trivial?
For example how would you proof that $$ \int_{-\infty}^\infty \frac{{\rm e}^{ikt} \, {\rm d}k}{\left(1+a^2\sin^2(k)\right)\left(k-b\right)^{\epsilon}} $$ where $\Im (b) < 0$, $0<\epsilon\leq 1$, $a>0$, converges and for which $t$. Analogously one could ask when does $$ \sum_{n=-\infty}^\infty \frac{ {\rm e}^{i\pi tn} }{ \left( n + c \right)^{\epsilon} } $$ converge $(\Im(c)>0)$? Feel free to try this.
A nice book in my opinion is "Advanced Mathematical Methods with Maple", by D. Richards. It is used as a text book in the homonymous course in the MSc in mathematics by the Open University. It is not only about convergence, but it has quite a theoretical underpinning and practical exercises too about that topic, where maple is used to illustrate and investigate the subject.