Series Convergence $\sum_{n=1}^{\infty} \frac{1}{n} \left(\frac{2n+2}{2n+4}\right)^n$ I have to show if this series converges or diverges. I tried using asymptotics, but it's not formally correct as they should work only when the arguments are extremely small. Any ideas?
$$
\sum_{n=1}^{\infty} \frac{1}{n} \left(\frac{2n+2}{2n+4}\right)^n
$$

Without any additional algebra, just notice that after cancellations both expressions
$$
\frac{\bigg(1+\frac{1}{2n}\bigg)^n}{\bigg(1+\frac{2}{n}\bigg)^n} = \frac{O(1)}{O(1)}=O(1)
$$
Hence your summand is $f_n = \frac{1}{n} \cdot O(1)$.
