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Math Help - sequences and series of functions

  1. #1
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    Exclamation sequences and series of functions

    let (gn) be a sequence of functions on D such that gn+1(x)< gn(x) for all x in D. if (gn)-> 0 uniformly on D, show that Σ (-1)^n(gn) converges uniformly on D
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by lavender87 View Post
    let (gn) be a sequence of functions on D such that gn+1(x)< gn(x) for all x in D. if (gn)-> 0 uniformly on D, show that Σ (-1)^n(gn) converges uniformly on D
    The series is pointwise convergence by the alternating series test (since the series is decreasing all the terms are non-negative on D etc).

    The conditions imply that there exists a decreasing positive sequence a_n \to 0 as n\to \infty such that g_n(x)\le a_n for all x \in D.

    Then (since the remainder for a truncated alternating series is bounded by the absolute value of the first neglected term):

     \left| \left(\sum_0^{\infty} (-1)^n g_n(x)\right) - \left(\sum_0^n (-1)^n g_n(x)\right) \right|\le g_{n+1}(x) \le a_{n+1}

    etc.

    CB
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