Applications of the infinite series of Cauchy residue theorem and its extensions

In this paper, the general concept of residue and general residue formulas are introduced. The general residue provides a method to study functions with infinite isolated singularities and the non-isolated singularities, which are the limits of sequences of isolated singularities. The general concept and residue theorem cover the shortage of the classical Cauchy residue theorem which can only deal with finite many isolated singularities. Then one uses the extended residue theorem on non-isolated singularities to calculate some infinite series. One can compute some classical series coming from different branches of mathematics, such as elliptic functions, number theory, and other examples. Using this method, one can see that it is a very useful method to deal with absolutely convergent series. More excitingly, this way is easy to use unlike most other methods for series. Thus, one can image that it has amazing applications in complex analysis and even other branches.