# Is this uniformly convergent?

1. Jan 9, 2013

### Artusartos

1. The problem statement, all variables and given/known data

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]?

2. Relevant equations

3. 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?

2. Jan 9, 2013

### micromass

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

3. Jan 9, 2013

### 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 $y = 1/n$?