Determine whether or not the series $\sum\limits_{n=2}^\infty (\frac{n+4}{n+8})^n$ converges or diverges Does the series converge or diverge?
The series $\sum\limits_{n=2}^\infty (\frac{n+4}{n+8})^n$
I tried the root test to get rid of the nth power but the limit equaled $1$ so the test is inconclusive. How else can I determine if this series converges or diverges?

$\begin{array}\\
a_n
&=\left(\dfrac{n+u}{n+v}\right)^n\\
&=\left(\dfrac{n+v-v+u}{n+v}\right)^n\\
&=\left(1+\dfrac{-v+u}{n+v}\right)^n\\
&\to e^{u-v}\\
\end{array}
$
since
$(1+\frac{a}{x})^x
\to e^a
$.
