To test whether $\sum_{n=1}^\infty\frac{n+2}{2^n+3}\sin\left[(n+\frac12)\pi\right]$ converges To determine whether the following sequence converges or divergence
$$\sum_{n=1}^\infty\frac{n+2}{2^n+3}\sin\left[(n+\frac12)\pi\right]$$
I don't know which test to use here, but my guess is it may be a comparison test but how to determine which series to use?

Hint: I assume you mean $$\sin\left[(n+1/2)\pi\right]=(-1)^{n}$$
By using this you can see that this is an alternating series. Use the Leibniz criterion to rule out convergence.
