Proving uniform convergence of $x^n-x^{2n}$

Question

I need help checking if the series of function {$x^n-x^{2n}$} on two cases:

• when $0\le x\le 1$

• when $\frac{1}{3}\le x \le \frac{1}{2}$

I managed to prove that in the first case:

the limit function is $f(x)=0$ so $r(x)=x^n-x^{2n}$ I have a maximum point when $x=\frac{1}{2}^{\frac{1}{n}}$ so $lim_{n\rightarrow \infty}r(x)=\frac{1}{4}\ne 0$

my question is why Dini's theorem does not apply on this case.

for the second case:

I don't have a maximum point so how do I prove it uniformly converges?

any insight will be very helpful.

Answer 1

Dini's theorem does not apply because the sequence is not monotone.

For the second case: $f'(x)=nx^{n-1}+2nx^{2n-1}=nx^{n-1}(1-2x^n)$, which is greater than $0$ in $\left[\frac13,\frac12\right]$. So, you do have a maximum: $\left(\frac12\right)^n-\left(\frac12\right)^{2n}=\left(\frac12\right)^n\left(1-\left(\frac12\right)^n\right)$. Since the limit of this sequence is $0$, your sequence converges uniformly to the null function.