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Thread: 2 calculus quesions

  1. #1
    Dec 2009

    2 calculus quesions

    Hey there,

    I'll be glad to get some help in the following questions:

    1. Let g(x) be a differentiable function at (0,1] which satisfies:  x^{2}g(x) is a monotonic ascending function at
    (0,1] and:  lim_{x \to 0^{+}} x^{2}g(x)=0 , g'(x) is continous at (0,1].
    Check whether the integral  \int_{0}^{1} g(x)sin(\frac{1}{x})dx converges.

    2. Let f(x) be a function defined by:  f(x)=\Sigma_{n=1}{\infty}a_{n}sin(nx) .
    Prove that if the series  \Sigma_{n=1}^{\infty}n|a_{n}| converges then f(x) is differentiable for every real x.

    Thanks in advance
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  2. #2
    MHF Contributor chisigma's Avatar
    Mar 2009
    near Piacenza (Italy)
    Setting \frac{1}{x}=t the integral becomes...

    \int_{0}^{1} g(x)\cdot \sin \frac{1}{x}\cdot dx = \int_{1}^{\infty} g(\frac{1}{t})\cdot \frac{\sin t}{t^{2}}\cdot dt (1)

    Now the function...

    \gamma (t) = \frac{g(\frac{1}{t})}{t^{2}} (2)

    ... is monotonically decreasing for t>1 nd ...

    \lim_{t \rightarrow \infty} \frac{g(\frac{1}{t})}{t^{2}}=0 (3)

    ... so that is...

    |\int_{1}^{\infty} g(\frac{1}{t})\cdot \frac{\sin t}{t^{2}}\cdot dt| < |\int_{1}^{\pi} \gamma(t)\cdot \sin t\cdot dt| + \pi \cdot  |\sum_{n=1}^{\infty} (-1)^{n} \gamma (n\cdot \pi)| (4)

    ... and the integral converges...

    Kind regards

    \chi \sigma
    Last edited by chisigma; Apr 18th 2010 at 05:15 AM. Reason: Added a 'pi' in (4)...
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  3. #3
    Dec 2009
    Thanks a lot
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