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Thread: uniform convergence

  1. #1
    Junior Member sweet's Avatar
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    uniform convergence

    if we have

    sum{n=0/to infinit} f_n (x)

    such that n in N and x in [-1.1]

    f_n =x/(1+x^2)^n


    r this series confergence?is it uniform convergence?

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  2. #2
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    Quote Originally Posted by sweet
    if we have

    sum{n=0/to infinit} f_n (x)

    such that n in N and x in [-1.1]

    f_n =x/(1+x^2)^n


    r this series confergence?is it uniform convergence?

    Ya have,
    $\displaystyle \sum^{\infty}_{k=0}\frac{x}{(1+x^2)^k}$
    Thus, $\displaystyle a_{k+1}=\frac{x}{(1+x^2)^{k+1}}$ and $\displaystyle a_k=\frac{x}{(1+x^2)^k}$ apply the ratio test,
    $\displaystyle \lim_{k\to \infty}\frac{x}{(1+x^2)^{k+1}}\cdot\frac{(1+x^2)^k }{x}$
    This gives,
    $\displaystyle L=\frac{1}{1+x^2}$
    It converges absolutely when,
    $\displaystyle L<1$
    Thus,
    $\displaystyle 1+x^2>1$
    Thus, $\displaystyle x>0$.

    It diverges when,
    $\displaystyle L>1$
    thus,
    $\displaystyle 1+x^2<1$
    WHich is impossible.

    It is inconclusive when, $\displaystyle L=0$,
    Thus, $\displaystyle x=0$.
    But we can this that this series is convergent abolsutely thus, this is absolutely convergenet everywhere.
    Last edited by ThePerfectHacker; May 24th 2006 at 01:49 PM.
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  3. #3
    Junior Member sweet's Avatar
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    ok now we know it's convergence but what about uniform convergence
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