Prove that it converges uniformly

[fabiancillo]

Hello, I have problems with this exercise:

Prove that the sequence $f_n : [0,1] \longrightarrow{\mathbb{R}}$ defined by $f_n(t)=t^n(1-t)$ converges uniformly to the null function in [0,1]Thanks

 

[Evgeny.Makarov]

Prove that $\max_{t\in[0,1]}f_n(t)\to0$ when $n\to\infty$.

 

[soloenergy]

for reaching out about this exercise. It looks like you are trying to prove the uniform convergence of a sequence of functions. To do this, you will need to show that for any given $\epsilon>0$, there exists an $N \in \mathbb{N}$ such that $|f_n(t)|<\epsilon$ for all $n \geq N$ and for all $t \in [0,1]$.

One approach to proving this is to use the Weierstrass M-test. This theorem states that if there exists a sequence of positive numbers $M_n$ such that $|f_n(t)| \leq M_n$ for all $n$ and for all $t \in [0,1]$, and if $\sum_{n=1}^{\infty} M_n$ converges, then the series $\sum_{n=1}^{\infty} f_n(t)$ converges uniformly on $[0,1]$.

In this case, we can choose $M_n = 1$ for all $n$, since $|f_n(t)| \leq 1$ for all $t \in [0,1]$. And since $\sum_{n=1}^{\infty} 1$ is a convergent series, by the Weierstrass M-test, we can conclude that the sequence $f_n$ converges uniformly to the null function on $[0,1]$.

I hope this helps with your exercise. Let me know if you have any other questions or if you need further clarification on any of the steps. Good luck!