<s>[INST] To test the convergence of series 2 To test the convergence of series 2
$\displaystyle \frac{a+x}{1!}+\frac{(a+2x)^2}{2!}\frac{(a+3x)^3}{3!}+...\infty$
My Attempt: 
$$\begin{align}
\frac{u_n}{u_{n+1}} & = \frac{(a+nx)^n}{n!}\frac{(n+1)!}{(a+(n+1)x)^{n+1}} \\
 & = \frac{(a+nx)^n}{1}\frac{n+1}{(a+nx+x)^{n+1}} \\ 
 & = \frac{(x+a/n)^n}{1}\frac{1+1/n}{(x+a/n+x/n)^{n+1}}  \\
 & = \frac{x^n(1+a/nx)^n}{1}\frac{1+1/n}{x^{n+1}(1+a/nx+1/n)^{n+1}}  \\ 
 & = \frac{(1+a/nx)^n}{1}\frac{1+1/n}{(1+a/nx+1/n)^{n+1}}\frac{1}{x} \\
 & = ??
\end{align}$$
I see an "e" in the limit $n\to \infty$ here, but can't quite get "there". Any hints?
 [/INST] Follow Did's advice, you should get something like 
$$
a^{\frac{1}{n}}_n = \frac{a+nx}{(n!)^{\frac{1}{n}}} \sim \frac{e(a+nx)}{n} (2 \pi n)^{-\frac{1}{2n}}
$$
The last term converges to 1, so the series converges for $x<e^{-1}$
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