Prove that $$\prod_{n=1}^{\infty}\left(\frac{2n+3}{2n+2}\right)$$ converges or diverges.
Taking analysis this year has really intrigued me. We do not cover this topic in analysis, but I did some research on my own after learning about conditionally convergent sequences. I have become interested in learning about infinite products. From my research, it seems that a criteria of a convergent infinite product is that the sequence converges to 1. This sequence does converge to 1 so it meets this criteria. I believe I need to show that
$$\sum_{n=1}^{\infty} \ln\left(\frac{2n+3}{2n+2}\right)$$
converges. I tried to prove this using the ratio test, but I got nowhere with that. Any suggestions?