I haven't read all the way through this yet, but I have read through your proof of your test for series convergence.

In the test of convergence your statement of the test and notation needs to be improved.

It could read something like:

Also I cannot follow your argument here:Test of Convergence

The series converges if:

Please inprove the notation and if possible clarify the reasoning.

(I will leave you post here for two weeks for you to prot a revision after that I will removing it as the standard of presentation is at present not acceptable)

CB