I know that this set:
$$\{i\ |\ \ \phi_i(n) \text{ converges } \}$$
is not recursive and that this can be shown by Rice's theorem.
But everywhere I look i just found that it's not recursive because of Rice's theorem or because it's not a trivial property. Can anyone provide any steps for the application of Rice's theorem for this function or in general to any other function.
Thank you
Rice's theorem says that if $F$ is a subset of the partial recursive functions such that $F$ is neither empty nor contains all partial recursive functions (that is, it is nontrivial), then $\{e\ |\ \phi_e \in F\}$ is not recursive. In particular this $F$ is generally determined by some property of partial recursive functions; in your example, $F$ is the set of partial recursive functions which converge on input $n$ (that is, have $n$ in their domain).
Since there are some partial recursive functions which converge on input $n$ (e.g., any total recursive function) and some which do not (e.g., the function which converges nowhere), this set $F$ is neither empty nor the whole set of partial recursive functions. That is, the hypothesis of Rice's theorem holds and therefore $\{i\ |\ \phi_i(n)\text{ converges}\}$ is not recursive.