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Convergence, differentiable, integrable, sequence of functions

  1. Apr 16, 2010 #1
    1. The problem statement, all variables and given/known data

    For [tex]k = 1,2,\ldots[/tex] define [tex]f_k : \mathbb{R} \to \mathbb{R}[/tex]
    by [tex]f_k(x) = \sqrt{k} x^k (1 - x)[/tex]. Does [tex]\{ f_k \}[/tex] 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. jcsd
  3. Apr 16, 2010 #2

    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.
  4. Apr 16, 2010 #3
    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?
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