Suppose we have an alternating series of the form $$\sum_{n=2}^{\infty} \frac{(-1)^{f(n)}}{n^2} \text{ where } f(n) = \begin{cases}1 & \text{n is prime} \\ 0 & \text{otherwise}\end{cases}$$.
How would one go about proving that this series converges or not? Does this series have a closed form solution?