Does the series converge or diverge? Did I do this correctly? Does the following series converge or diverge? 
$\sum\limits_{j=1}^\infty \frac{(1+\frac{1}{j})^{2j}}{e^j}$
Using the ratio test, I found $|\frac{a_{j+1}}{a_j}|$ to be $|\frac{(1+\frac{1}{j+1})^{2j+2}}{e(1+\frac{1}{j})^{2j}}|$
To find the lim sup of this, I looked at the ratio as $j \rightarrow \infty$, and arrived at $\frac{1}{e}<1$, so by the ratio test, the series would converge absolutely.
Does this look valid? I wasn't sure about the step of taking the limit as $j \rightarrow \infty$ to find the lim sup.
Thank you!

You could also say this: the numerator goes to $e^2$ from below, so the series is less than $\sum \dfrac{e^2}{e^j}$, which converges by geometric test.
