I have a sum $\displaystyle \displaystyle{ \sum_{k=0}^\infty \frac{1}{k!}\left\{\int_0^1 (x\ln x)^k dx\right\} }$.

Now,per partesit "is easy to show that" $\displaystyle \displaystyle{ \int_0^1 (x\ln x)^k dx=\frac{(-1)^kk!}{(k+1)^{k+1}} }$ (for this I should use $\displaystyle \displaystyle{ \lim_{x\to 0}x\ln x=0 }$).

I "know" that the $\displaystyle \lim$ holds ... but how to use it inper partesintegration?