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?
Best Answer
Guide:
We have $$\sum_{n=1}^\infty \sin (\frac{n\pi}{2}) \frac{n^2+2}{n^3 +n}= \sum_{n=1}^\infty(-1)^{n+1} \frac{(2n-1)^2+2}{(2n-1)^3 +(2n-1)}$$
Try alternating series test.