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Math Help - bounded variation

  1. #1
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    bounded variation

    Let f be a funtion from R^+ to R such that f is of bounded variation on [0,n] where n \in Z^+. Let V_f (0,n) be the total variation of f on [0,n]. In addition, f satisfies \mid f(n)-f(0)\mid \leq M_n and \sum_{n=1}^\infty M_n \leq \infty.
    Determine whether or not lim_{n\rightarrow\infty} V_f (0,n) exists.
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  2. #2
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    Quote Originally Posted by Kat-M View Post
    Let f be a funtion from R^+ to R such that f is of bounded variation on [0,n] where n \in Z^+. Let V_f (0,n) be the total variation of f on [0,n]. In addition, f satisfies \mid f(n)-f(0)\mid \leq M_n and \sum_{n=1}^\infty M_n \leq \infty.
    Determine whether or not lim_{n\rightarrow\infty} V_f (0,n) exists.
    the limit is not necessarily finite. for example f(x)=\sin(\pi x) satisfies all the conditions in your problem but V_f(0,n)=\int_0^n |f'(x)| \ dx = \int_0^{n \pi} |\cos x| \ dx=2n.
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