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Thread: Sequence of function 2

  1. #1
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    Sequence of function 2

    Let$\displaystyle (f_n)^\infty_{n=1}$ a sequence of function which defined recursively on interval $\displaystyle [a,b] $:

    $\displaystyle f_n(x)=\sqrt{xf_{n-1}(x)}$, $\displaystyle f_0(x)\equiv1$

    1. Prove that the sequence $\displaystyle (f_n)^\infty_{n=1}$ converges on interval $\displaystyle [a,b] $ to limit continuous function$\displaystyle f$.

    2. Prove that the convergence is uniform.
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