The title could be: Uniform Convergence of f_n(x) = nx^n(1-x) on [0,1]?

[Artusartos]

Homework Statement

Consider f_n(x) = nx^n(1-x) for x in [0,1].

a) What is the limit of f_n(x)?

b) Does f_n \rightarrow f uniformly on [0,1]?

Homework Equations

The Attempt at a Solution

a) 0

b) Yes...

We know that sup|f_n(x) - f(x)| = |n{\frac{1}{2}}^n(1-\frac{1}{2})|...

and

lim_{n \rightarrow \infty} [sup\{ |f_n(x) - f(x)|: x \in [0,1]\}] = 0

So it must be uniformly convergent on [0,1].

Do you think my answer is correct?Thanks in advance

 

[micromass]

Homework Statement

Consider f_n(x) = nx^n(1-x) for x in [0,1].

a) What is the limit of f_n(x)?

b) Does f_n \rightarrow f uniformly on [0,1]?

Homework Equations

The Attempt at a Solution

a) 0

b) Yes...

We know that sup|f_n(x) - f(x)| = |n{\frac{1}{2}}^n(1-\frac{1}{2})|...

Why? How do you know they will always obtain their maximum in 1/2??

 

[jbunniii]

If we let y = 1-x, then we may write
f_n(1-y) = n y (1 - y)^n
Now what happens if you choose y = 1/n?