Series convergence: $\sin (n \frac{\pi}{2})$ 
Determine whether the following series :
  $$\sum_{n=1}^\infty \sin \left(\frac{n\pi}{2}\right) \frac{n^2+2}{n^3 +n}$$
  converges absolutely, conditionally or diverges.

I know that for even natural numbers the expression will equal zero and that for odd values of $n$ the value of $\sin$ will go from $1$ to $-1$.
Could I theoretically reduce this series into a subseries:
$$\sum_{n=0}^\infty \sin \left(\frac{(2n+1)\pi}{2}\right) \frac{(2n+1)^2+2}{(2n+1)^3 +2n + 1}$$
And then treat it as if it were a standard alternating series?

HINT


*

*Note that 
$$
\frac{n^2+2}{n^3+n}
 = \frac{1}{n} \times \frac{n^2+2}{n^2+1}
 = \frac{1}{n} \left[ 1 + \frac{1}{n^2+1} \right]
 = \Theta(1/n)
$$
Does this series converge?

*Adding $\sin(n\pi/2)$ in the front effectively kills all the even-$n$ terms and makes an alternating series out of the odd ones -- does the alternating series converge?

