\(\displaystyle \Sigma_{n=1}^{\infty}\frac{2\cdot 3\cdot 4\cdot\cdot\cdot(n+1)x^n}{n!}\)

\(\displaystyle \lim_{n->\infty}\mid\frac{u_{n+1}}{u_n}\mid=\lim_{n->\infty}\mid\frac{2\cdot 3\cdot 4\cdot\cdot\cdot(n+2)x^{n+1}}{(n+1)!}\frac{n!}{2\cdot 3\cdot 4\cdot\cdot\cdot(n+1)x^n}\mid\)

Using \(\displaystyle (n+1)!=n!(n+1)\), this simplifies:

\(\displaystyle =\lim_{n-\infty}\mid\frac{(n+2)x}{(n+1)^2}\mid\)

According to my solution manual, this is incorrect. The limit \(\displaystyle \lim_{n->\infty}\mid\frac{u_{n+1}}{u_n}\mid=\lim_{n-\infty}\mid\frac{(n+2)x}{(n+1)}\mid\)

Where am I going wrong?