I have the following sequences of functions:
$f_n(x)=n^3x^n(1-x)$ on $[0,1/2]$ (conjecture: this converges uniform to $0$)
$f_n(x)=\frac{x}{nx+1}$ on $(0,1)$ (conjecture: also uniform limit to $0$)
$f_n(x)=\frac{nx^2+1}{nx+1}$ on $[1,2]$ (conjecture: pointwise limit to $f(x)=x$)
I think you can use Dini's Theorem in the first case. But how to do this? How to obtain the limits in the other cases?
Thank you.
The conjectures are true.
For the first problem note that for large $n$ we have $$ n^{3}x^{n}(1-x) = \frac{n^{3}}{\exp (n\log |x|)}(1-x) < \frac{n^{3}}{n^{4}}(1-x) < \frac{1}{n} $$ for all $x \in ]0,1/2]$.
For the second problem note that $$ \frac{x}{nx+1} < \frac{x}{nx} = \frac{1}{n} $$ for all $n\geq 1$ and all $x \in ]0,1[$.
For the last note that $$ \bigg| \frac{nx^{2}+1}{nx+1} - x \bigg| = \bigg| \frac{nx^{2}+1-nx^{2}-x}{nx+1} \bigg| = \bigg| \frac{1-x}{nx+1} \bigg| \leq \frac{1}{nx+1} + \frac{x}{nx+1} \to 0 $$ for all $x \in [1,2]$.