I posted similiar question an hour ago, so I just want to check, if I undestand it correctly for another problem.
I am supposed to determine if series of function $\left \{ n^{2}x^{n}\left ( 1-x \right ) \right \}_{n=1}^{\infty}$ on interval $M\left [ 0,1 \right ]$is converging pointwise and uniform.
Pointwise:
Yes, it is, because $\lim_{n \to \infty }( n^{2}x^{n}\left ( 1-x \right ))=0 $
Uniform:
$a_{n}=sup_{x \in \mathbb{R} }\left | n^{2}x^{n}\left ( 1-x \right )-0 \right |=\left |n^{2}x^{n}\left ( 1-x \right ) \right |$
Maximum of function is $x=\frac{n}{1+n}$ so:
$\lim_{n \to \infty }(n^{2}\left ( \frac{n}{1+n} \right )^{n}\left ( \frac{1}{1+n} \right ))= \infty$, therefore function does not converge uniform.
Is that correct?
Almost. You should have stated that the maximum of $x\mapsto n^2x^n(1-x)$ is attained at $\frac n{n+1}$. The maximum itself is $n\left(\frac n{n+1}\right)^{n+1}$. The rest is fine.