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Math Help - limit question (indeterminate forms??)

  1. #1
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    limit question (indeterminate forms??)

    Question:
    We work in the real numbers. Are the following true or false? Give a proof or counterexample.
    (a) If \sum a^4_n converges, then \sum a^5_n converges.
    (b) If \sum a^5_n converges, then \sum a^6_n converges.
    (c) If a_n \geq 0 for all n, and \sum a_n converges, then na_n \rightarrow 0 as n \rightarrow \infty.
    (d) If a_n \geq 0, for all n, and \sum a_n converges, then n(a_n - a_{n-1}) \rightarrow 0 as n \rightarrow \infty.
    (e) If a_n is a decreasing sequence of positive numbers, and \sum a_n converges, then na_n \rightarrow 0 as n \rightarrow \infty.

    MY WORK:
    (a) and (b) can be proved similarly. Since \sum a_n^4 converges, for some N, when n \geq N, then a_n^4 < 1. Take \beta s.t., a_n^4 < \beta < 1. That is, a_n < (\beta)^{1/4} < 1. Also, |(\beta)^{1/5}| < 1. This implies that |a_n^5| < 1 and therefore \sum |a_n^5| converges.

    (c) This is where I get confused. THis seems like an indeterminate form, we have never done this in class. same for (d) and (e). any suggestions?
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  2. #2
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    Your work on part a) could a bit smoother: (a_n)^4<1 implies
    |a_n|^5\le (a_n)^4<1.

    For part b) consider a_n  = \dfrac{{\left( { - 1} \right)^n }}{{\sqrt[{10}]{n}}}.
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  3. #3
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    Here is part e).
    For ease of notation use S_N  = \sum\limits_{k = 1}^N {a_k } for the partial sums.
    Because the series converges , the sequence (S_n) is Cauchy.
    If \varepsilon  > 0 there is positive integer K such if j>K then S_j-S_K<\frac{\varepsilon }{2}.
    We also know that (a_n)\to 0. So there is a positive integer J>K such that n \geqslant J\, \Rightarrow \,a_n  < \frac{\varepsilon }{{2K}}.

    Now we have setup the machinery, we just have to find the tricks.
    If p\ge J then pa_p=(p-K)a_p+Ka_p.
    But we notice that (p-K)a_p\le S_p-S_K.

    Can you finish?
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