proof of convergence for the polygon circumscribing constant $\sum _{3} ^{\infty} \ln(\sec(\pi/ n))$

The polygon circumscribing constant is found by: $$\prod _3 ^\infty \sec \left( \frac\pi n \right)$$

I am trying to find a proof that this product converges. I know it is equal to:

$$\exp \left( \sum _{3} ^{\infty} \ln\left(\sec\left(\frac\pi n\right)\right) \right)$$

So I just need to show that sum converges. I do not see an easy way to use any of the convergence tests. In particular, I spent way to long trying to integrate this function to no avail.

So what convergence test is useable in this case?