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Thread: differentiability and continuity of function

  1. #1
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    differentiability and continuity of function

    Prove that the function $\displaystyle f(x)=\sum^{\infty}_{n=1} |\sin x|^{\sqrt{n}}$ is continuous on $\displaystyle (-1,1)$. Examine differentiability of it on $\displaystyle (-1,1)$.
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  2. #2
    Super Member girdav's Avatar
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    Re: differentiability and continuity of function

    What did you try? Did you already show the convergence of the series?
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  3. #3
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    Re: differentiability and continuity of function

    Because of $\displaystyle |\sin x|<\sin 1<1$ on $\displaystyle (-1,1)$we know that the series is convergent. Is that right?
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  4. #4
    Super Member girdav's Avatar
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    Re: differentiability and continuity of function

    Yes (maybe you should detail the fact that the series $\displaystyle \sum_{n=0}^{+\infty}a^{\sqrt n}$ converges for $\displaystyle 0<a<1$). Use the fact that the series is normally convergent. For the differentiability, look at $\displaystyle 0$.
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