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Math Help - cauchy

  1. #1
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    cauchy

    given that {a_n} is a sequence of numbers that are greater than 0 and that sum a_n diverges. show that sum a_n / (1+ a_n) diverges.

    how do i show that?

    i know that a_n / (1+ a_n) = 1- 1/ (1+ a_n) but what else?
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  2. #2
    Senior Member roninpro's Avatar
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    We can break the problem into two cases.

    Case 1: Suppose that a_n\not \to 0. Then,

    \displaystyle \sum \frac{a_n}{1+a_n}=\sum \left(1-\frac{1}{1+a_n}\right)

    But then the terms of the sum on the right do not approach zero either, so the sum is divergent.

    Case 2: Suppose that a_n\not \to 0. Try using the limit comparison test.
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