Garrett Black

2022-06-21

How do I check whether the sequence converges uniformly?
${f}_{n}={x}^{n}-{x}^{n+1}={x}^{n}\left(1-x\right),\phantom{\rule{1em}{0ex}}x\in \left[0,1\right]$

Bruno Hughes

It is controlled by
$\begin{array}{r}{\left(1-\frac{1}{n+1}\right)}^{n}\left(1-\frac{n}{n+1}\right)={\left(1-\frac{1}{n+1}\right)}^{n}\frac{1}{n+1}.\end{array}$
We know that
$\begin{array}{r}{\left(1-\frac{1}{n+1}\right)}^{n}\to {e}^{-1},\end{array}$
so for large n,
which tends to zero as $n\to \mathrm{\infty }$

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