To test whether $\sum_{n=1}^\infty\frac{n+2}{2^n+3}\sin\left[(n+\frac12)\pi\right]$ converges

Question

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?

Answer 1

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.

Answer 2

The absolute convergence and hence the convergence follows easily using asymptotic comparison

$$\left\vert\frac{n+2}{2^n+3}\sin\left[(n+\frac12)\pi\right]\right\vert=\frac{n+2}{2^n+3}=o\left(\frac1{n^2}\right), \text{where} \;n\to \infty$$