This is the explanation to the answer to the question: Test the convergence of the series $\displaystyle\sum\limits_{n=1}^ \infty \frac{n!\, \pi^n}{e^{n^2}}$
I don't understand what they did with the factorial terms. How did they disappear? It seems to me that $(n+1)!$ is much larger than $n!$ - which will change the limit to approach infinity. Can someone help?
This is simply because, by definition, $$n!=1\cdot2\cdot3\dotsm(n-1)\cdot n $$ and therefore, $$\frac{(n+1)!}{n!}=\frac{\color{red}{1\cdot2\cdot3\dotsm(n-1)\cdot n}\cdot(n+1)}{\color{red}{1\cdot2\cdot3\dotsm(n-1)\cdot n}}=n+1$$ since the red terms cancel out exactly.
In some sense your intuition is right, in absolute terms $|(n+1)!-n!|$ is very large and does grow to infinity rapidly. However, here we have the ratio $(n+1)!/n!=n+1$, which still grows to infinity, but in a relatively tame way---it is only linear. The limit as $n\to\infty$ happens to be $0$ here because although we have a factor that grows linearly to infinity, the $e^{2n+1}$ factor in the denominator is much more significant (as it is an exponential term), and hence dominates the linear term and forces the expression overall to tend to $0$. This is perhaps a nice illustration of why not to trust your initial instincts before you've worked out the algebra and can confidently see what's going on.