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Math Help - Rearrangement of Terms

  1. #1
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    Rearrangement of Terms

    Let sigma Un be a convergent series, and let sigma Vn be a rearrangement of it. In the rearrangement, suppose that no term of the original series is moved more than N places from its original position, where N is a fixed number. Show that the new series is convergent and has the same value as the old one.

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  2. #2
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    This question seems trivial... Since addition is associative and commutative it shouldn't matter what order the terms are in...
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  3. #3
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    I think the question is about infinite series. From Wikipedia: "Bernhard Riemann proved that a conditionally convergent series may be rearranged to converge to any sum at all, including ∞ or −∞."
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    Let a=\sum_{n=1}^\infty U_n. Further, let M be such that for S_1=\sum_{n=1}^M we have \lvert S_1-a\rvert<\varepsilon/2. What can be said about S_2=\sum_{n=1}^{M+N} U_n? (Imagine that M\gg N.) All terms in S_1 are also in S_2. Also, N terms from the "tail" (i.e., remainder) of S_1 are in S_2, but the whole tail of S_1 is small, i.e., <\varepsilon/2. Therefore, S_2 is close to a.
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