Properties of $\sum\limits_{n\ge2}\frac{n(n-1)x^n}{n!}$ for $x\in \mathbb{R}$

Question

This is from an MCQ Contest.

for all integer $n$ greater than or equal to $2$ Let $$\forall n\geq 2\qquad u_n=\dfrac{n(n-1)x^n}{n!}, \qquad x\in \mathbb{R}$$

1. for all $x$ in $\mathbb{R}$, the series with general term $u_n$ is convergent and has the sum $x^{2}e^{x}$
2. the series with general term $u_n$ is convergent if and only if $|x|<1$
3. the series with general term $u_n$ is convergent if and only if si $|x|>1$
4. for all $x$ in $\mathbb{R}^{*}$, the series with general term $u_n$ is convergent if and only if $|x|=1$

My thoughts

1. True

it's true that $u_n$ convergent since the ratio test gives us : $$|\dfrac{u_{n+1}}{u_n}|=\dfrac{|x|}{n+1}\to 0 < 1$$ then the series converges absolutely beside its sum $$\sum_{n=2}^\infty \frac{n(n-1)x^n}{n!}=\sum_{n=2}^\infty \frac{x^n}{(n-2)!}=\sum_{n=0}^\infty \frac{x^{n+2}}{n!}=x^2\sum_{n=0}^\infty \frac{x^n}{n!} $$

• Since $\forall x \in \mathbb{R}\ \sum u_{n}$ is convergent so implies that 2 3 and 4 are False because it convergent for all $\forall x \in \mathbb{R}$

• Is my proof correct