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

    2) S_n(x)= nx/(x+n).

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

    4) 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 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 S(x)=\lim_{n\to +\infty}S_n(x) ?
    S(x)=\lim_{n\to +\infty}S_n(x) actually, i'm not sure what is the difference between 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
    S(x)=\lim_{n\to +\infty}S_n(x) actually, i'm not sure what is the difference between S(x) and S_n(x)

    For example 2):

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

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