test for convergence or divergence

Question

So I am looking at the following series:

$$\sum_{n=1}^{\infty}{{ln(n)+n^p+r^n+n!}\over{n^n-n!-r^n-n^p-ln(n)}}$$

Before testing, I wanted to look at some series that I can compare this to but haven't been able to think of one yet. Any suggestion?

Answer 1

Hint: If $r>1$, $$\ln n=o(n^p),\enspace n^p=o(r^n),\enspace r^n=o(n!),\enspace n!=o(n^n),$$ hence the numerator is equivalent at $+\infty$ to $n!$, and the denominator to $n^n$.

Then use Stirling's formula, and apply the root test.