I have the problem that I could not see my fails in the my calculations and why does exists just one right way of showing the convergence or divergence of that formula. the first version is the version of the tutor and the version A and B are my versions. there I get 2 different kinds of results what I don't know why. Could someone take a look about that and tell me where are my fails and maybe could someone explain me why I could not use my forms of results with the imaginary parts in it? thx
When testing for convergence, as many initial terms as you want can be ignored. Therefore, you need only consider terms with $n > 2$.
In your answer A, you should be adding the $-2n$ term, not multiplying.
Similarly, in your answer B, you should add the terms so that $\ln(((n-2)^3)e^{-n(x+2)}) =3\ln(n-2)-n(x+2) $.