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?
$$
\sum_{n=1}^{\infty} \frac{1}{n} \left(\frac{2n+2}{2n+4}\right)^n
$$

$\frac{2n+2}{2n+4} = \frac{n+1}{n+2}$
With $\frac{n+1}{n+2} \le  1+\frac1n$ and $\lim_{n\to \infty} (1+\frac1n)^n = e$ you can use comparison test to show that it is divergent.
