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Math Help - Example of sequence of functions

  1. #1
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    Example of sequence of functions

    Hello,

    Could you help me to solve http://i51.tinypic.com/2nhhiqp.png
    Actually I am totally desperate of it and can't even start. I am not good in exaples such these ("give an example..."). It would be perfect if you type some steps how to find it
    (I hope I translated it right)

    Thanks a lot
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  2. #2
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    Hmm. I'm not sure if the interval <0,1> is supposed to denote (0,1) or [0,1]. If it's the former, just try nx^{n-1}. The latter is a bit trickier.
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  3. #3
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    I think it should be [0,1]. Could you explain each possible sequence please.
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    For the latter, choose

    f_n:[0,1]\to \mathbb{R}\quad,\quad f_n(x)=nx(1-x^2)^n
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  5. #5
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    Thank (could you describe how you found it?). One more thing - I hope i can join it to this one thread. We have:

    \lim_{n \to \infty} \int_{0}^{\infty} { \frac{cos (\frac{x}{n})} {(1 + \frac {x}{n})^{n}} }
    I solved it as: x\in [0,\infty)\:; { \frac{cos (\frac{x}{n})} {(1 + \frac {x}{n})^{n}} } \leq \frac{cos \, 0}{1^n} \:;\: x>0 (Weierstrass test - uniform convergence) so i can change integral with limit and then it is easy - result = 1

    Is this solution right ?
    Last edited by token22; May 1st 2011 at 09:45 AM.
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by token22 View Post
    Thank (could you describe how you found it?).

    It is a well known counterexample for

    \displaystyle\lim_{n \to{+}\infty}{\int_a^b f_n}=\int_a^b \displaystyle\lim_{n \to{+}\infty}{f_n}

    when we consider pointwise convergence.
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  7. #7
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by token22 View Post
    Is this solution right ?

    No, it isn't. Notice that for your bounds

    |f_n(x)|\leq M_n=1

    we have

    \displaystyle\sum_{n=1}^{+\infty}M_n\;\; \textrm{divergent}
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  8. #8
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    Then how can I prove that I can change integral with limit (it comes to me quite impossible to solve integral first anf then limit) ?
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  9. #9
    Behold, the power of SARDINES!
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    Quote Originally Posted by token22 View Post
    Then how can I prove that I can change integral with limit (it comes to me quite impossible to solve integral first anf then limit) ?
    Do you know Lebesgue's dominated convergence theorem?

    If so it will be very useful.
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  10. #10
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    I can't find the function that is greater and its integral do not divergent. Maybe I can't change integral with limit.
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