Convergence, differentiable, integrable, sequence of functions

[complexnumber]

Homework Statement

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?

Homework Equations

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?

 

[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.

 

[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?