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Thread: pointwise and uniform convergence problem

  1. #1
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    pointwise and uniform convergence problem

    Determine the convergence, both pointwise and uniform, on [0,1] for each of the following sequences:

    1) $\displaystyle S_n(x)= n^2x^2(1-cos1/[nx]), x not= 0; S_n(0)=0$.

    2) $\displaystyle S_n(x)= nx/(x+n)$.

    3) $\displaystyle S_n(x)= (1-cosnx)/nx, x not= 0; S_n(0)= 0$.

    4) $\displaystyle S_n(x)= nsin(x/n)$.


    I'm really struglling with these uniformly convergence problems, any help would be appreciated
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    For example, did you have any problems finding $\displaystyle S(x)=\lim_{n\to +\infty}S_n(x)$ ?
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  3. #3
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    Quote Originally Posted by FernandoRevilla View Post
    For example, did you have any problems finding $\displaystyle S(x)=\lim_{n\to +\infty}S_n(x)$ ?
    $\displaystyle S(x)=\lim_{n\to +\infty}S_n(x)$ actually, i'm not sure what is the difference between $\displaystyle S(x) and S_n(x)$
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by wopashui View Post
    $\displaystyle S(x)=\lim_{n\to +\infty}S_n(x)$ actually, i'm not sure what is the difference between $\displaystyle S(x) and S_n(x)$

    For example 2):

    $\displaystyle S(x)=\displaystyle\lim_{n\to +\infty}\dfrac{x}{(x/n)+1}=\dfrac{x}{0+1}=x$

    So, $\displaystyle S_n:[0,1]\to \mathbb{R}$ converges pointwise to $\displaystyle S:[0,1]\to \mathbb{R},\; S(x)=x$. Try to find $\displaystyle S(x)$ for the rest.
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