Does the following series converge or not?

Question

$$\sum_{i=1}^\infty \frac{n+\cos(n)}{n^2+n\log(n)+10\sin(n)}$$ I cannot think of any of my textbook methods to work with.

Answer 1

Note that for sufficiently large $n$ we have $$n+\cos n\ge {n\over 2}$$and $$n^2+n\log n+10\sin n\le 3n^2$$therefore$${n+\cos n\over n^2+n\log n+10\sin n}\ge{1\over 6n}$$which means that the series diverges.

Answer 2

It is just n/n^2 which diverges because you can take out all terms that are not n or n^2 because they are the greatest terms in the fraction as n approaches infinity. 1/n is the simplification and it diverges.