Answer to Question #169777 in Real Analysis for Aman Sharma

Question

Show that the sequence {𝑓𝑛} where 𝑓𝑛(𝑥) = 𝑛𝑥(1 − 𝑥)

𝑛 does not converge uniformly on [0, 1].

Answer 1

Show that the sequence {𝑓𝑛} where 𝑓𝑛(𝑥) = 𝑛𝑥(1 − 𝑥)^𝑛 

does not converge uniformly on [0, 1].

Solution. Find the limit function $f(x)=\lim\limits_{n\to\infty}f_n(x)=0,$ where $\lim\limits_{n\to\infty}nx(1-x)^n=\lim\limits_{n\to\infty}\frac{nx}{(1-x)^{-n}}=[\frac{\infty}{\infty}]=-\lim\limits_{n\to\infty}\frac{x(1-x)^{n}}{\ln(1-x)}=0$

is found by Lopital-Bernoulli rule. According to the criterion of uniform

convergence, functional sequence $f_n(x)=nx(1-x)^n$ does not

converge uniformly on [0,1]. Really, $\lim\limits_{n\to\infty}\sup\limits_{0\le x\le 1}|f(x)-f_n(x)|=$

$\lim\limits_{n\to\infty}\sup\limits_{0\le x\le 1}(nx(1-x)^n)=$ $\lim\limits_{n\to\infty}(1-\frac{1}{n+1})^{n+1}=\frac{1}{e}\ne 0.$

Answer: $f_n(x)=nx(1-x)^n$ does not converge uniformly on [0,1].