Can be $f_n:[0,1]\rightarrow\mathbb{R},\:f_{_n}(x)=x^n \cdot\ e^x$ uniform convergence? We have $f_n:[0,1]\rightarrow\mathbb{R},\:f_{_n}(x)=x^n \cdot\ e^x$.

I don't know how we can find the pointwise convergence...This sequence can be a uniform convergence? and explain your argument.
Pointwise convergence is very simple.

In order for a series of functions $f_n$ to converge pointwise towards a function $f$, the only condition is:


  For all values $x$, the limit $$\lim_{n\to\infty} f_n(x)$$ is equal to $f(x)$


Did you try to calculate the limit in your case? Since $x$ is fixed, it should be easy!
Pointwise convergence for any $x\in [0,1)$ we have $x^ne^x\to 0$ and $f_n(1)\to e$ so the sequence has a non continuous pointwise limit and therefore the sequence cannot be uniformly convergent