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Thread: how to show if they r uni. convergent

  1. #1
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    how to show if they r uni. convergent

    $\displaystyle f_{n}(x)=log(x+\frac{1}{n})
    f_{n}(x)=cos(\frac{x}{n}) $
    on (0,1)
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by silversand View Post
    $\displaystyle f_{n}(x)=log(x+\frac{1}{n})
    f_{n}(x)=cos(\frac{x}{n}) $
    on (0,1)
    I'm a little confused about what you have written.

    I assume those are two different functions?

    $\displaystyle f_n(x)=\log(x+\frac{1}{n})$ and $\displaystyle f_n(x)=\cos(\frac{x}{n})$

    Also, are you using $\displaystyle \log(x)$ to mean the same thing as $\displaystyle \ln(x)$?
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  3. #3
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    yes

    yes. they are two cases. and log means ln
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  4. #4
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by silversand View Post
    $\displaystyle f_{n}(x)=log(x+\frac{1}{n})
    f_{n}(x)=cos(\frac{x}{n}) $
    on (0,1)
    Second function: We know that this function converges to $\displaystyle \cos(0)=1$, the question is whether it does so uniformly. Using the difference-to-product identity, we have:

    $\displaystyle \left|\cos(0)-\cos\left(\frac{x}{n}\right)\right| = \left|-2\sin\left(\frac{x}{2n}\right)\sin\left(-\frac{x}{2n}\right)\right|\leq 2\left|\sin\left(\frac{x}{2n}\right)\right|$ because $\displaystyle |\sin(y)|\leq 1$ for all $\displaystyle y$.

    Since $\displaystyle 2\sin\left(\frac{x}{2n}\right)$ is increasing and positive on $\displaystyle (0,1)$, we know that $\displaystyle 2\left|\sin\left(\frac{x}{2n}\right)\right|\leq 2\sin\left(\frac{1}{2n}\right)$.

    It is now clear that for all $\displaystyle \epsilon>0$, we can choose $\displaystyle n$ large enough so that $\displaystyle \left|1-\cos\left(\frac{x}{n}\right)\right| \leq 2\sin\left(\frac{1}{2n}\right)<\epsilon$

    If you need an explicit formula, I think choosing $\displaystyle n>\frac{1}{2\sin^{-1}(\frac{\epsilon}{2})}$ will do the trick.
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