# Homework Help: Convergence, differentiable, integrable, sequence of functions

1. Apr 16, 2010

### complexnumber

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

For $$k = 1,2,\ldots$$ define $$f_k : \mathbb{R} \to \mathbb{R}$$
by $$f_k(x) = \sqrt{k} x^k (1 - x)$$. Does $$\{ f_k \}$$ converge? In
what sense? Is the limit integrable? Differentiable?

2. Relevant equations

3. The attempt at a solution

I don't know how to approach this question. How can I determine if the sequence converges? What are the theorems to dertermine if the limit is differentiable or integrable?

2. Apr 16, 2010

### snipez90

http://en.wikipedia.org/wiki/Uniform_convergence

In particular, note the difference between uniform convergence and pointwise convergence under Definition: Notes (this answers the convergence in what sense question). Look under Applications to see the theorems that guarantee that the limit function is differentiable or integrable.

3. Apr 16, 2010

### Hoblitz

Well, look at f_k carefully: it has exactly two real roots no matter what k is: zero and 1. What is happening outside of the interval [0,1]?

Inside the interval, it's a little more interesting. On the interval [0,1], for arbitrary k > 0 where does f_k achieve its maximum value, and what is that maximum value?