First it' not enough to show that `lim_(n->infty)a_n ne0`. For example series `sum1/n` is the case in which `lim_(n->infty)1/n=0` while `lim_(n->infty)sum_(n=1)^(infty)1/n=infty`
Meaning of `Theta.`
`f(x)=Theta(g(x))` means that `c_1g(x)leq f(x)leq c_2g(x)` for some`c_1,c_2 in RR`. To put it more simply it means that function `f` is behaving (growing and falling) at approximately the same rate (if we forget about constants) as function `g.`
For example: `3x^2=Theta(x^2)` because `4x^2 leq 3x^2 leq 2x^2`
For more on asymptotic notation see e.g. big O or consult any the literature: Knuth, Donald E.: The Art of Computer Programming vol.1 or Cormen, Leiserson, Rivest, Stein: Introduction to Algorithms
Case 1 - What hapens if `a_n` is divergent e.g. `a_n=n`
Case 2 - What hapens if `a_n` is convergent and is falling (or growing if negative) at the same rate as `1/n^p` for `p leq1`
Case 3 - This case is only here to fill up what happens if `p >1` but it does not actually interest us because in that case our series `sum a_n` is convergent what is in contradiction with our assumption.
I really hope this helpes.