For which $p$ the sequence $x^n$ converges in the Sobolev space $W^{1,p}(I)$?

Question

I would like to know for which $p$ the sequence $u(n)=x^n$ converges in the Sobolev space $W^{1,p}(I)$.

Is it true that converges only for $p=1$? I find out this looking for which $p$ the Sobolev norm of the sequence is finite. Am I right?

Thanks for the help

Answer 1

$$ \int_0^1 x^{pn} = \frac{1}{pn +1} x^{pn+1} |_0^1 <\infty =\frac{1}{pn+1}$$ and $$ \int_0^1 (n x^{n-1} )^p = \frac{n^p}{p(n-1)+1} $$ So for $1\leq p< \infty $, $x^n \in W^{1,p}$.

(1) But norm of $x^n$ goes to $\infty$ when $p> 1$. Hence in this case it does not converge

(2) Let $p=1$. The norm of $x^n$ goes to $1$. Note that the limit is $f(x)=0,\ x<1$ and $f(1)=1$, which is in $W^{1,p}(I)$. But $ \parallel f(x) \parallel_{1,p}= 0$. So $x^n$ does not converge in $W^{1,p}$.