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Math Help - real analysis

  1. #1
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    real analysis

    1. If ∑an with an >0 is convergent, then is ∑an always convergent? Either prove it or give a counterexample?
    2. If ∑an with an >0 is convergent, then is ∑(anan+1)^(1/2) always convergent? Either prove it or give a counterexample?
    3. Show that if a series is conditionally convergent, then the series obtained from its positive terms is divergent, and the series obtained from its negative terms is divergent.
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  2. #2
    Moo
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    Quote Originally Posted by yuan.69 View Post
    1. If ∑an with an >0 is convergent, then is ∑an always convergent? Either prove it or give a counterexample?
    If \sum a_n converges, it implies that \lim_{n \to \infty} a_n=0
    Hence, \forall \epsilon >0 ~,~ \exists N \in \mathbb{N} ~,~ \forall n \geqslant N, a_n<\epsilon
    now consider \epsilon=1 and consider that we are working on n \geqslant N
    then a_n^2 \leqslant a_n

    thus \sum_{n \geqslant N} a_n^2 \leqslant \sum_{n \geqslant N} a_n

    does this help ?
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  3. #3
    Senior Member JaneBennet's Avatar
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    Quote Originally Posted by yuan.69 View Post
    2. If ∑an with an >0 is convergent, then is ∑(anan+1)^(1/2) always convergent? Either prove it or give a counterexample?
    By AM–GM, \left(a_na_{n+1}\right)^{\frac12}\leqslant\frac{a_  n+a_{n+1}}2

    \therefore\ \sum_{n\,=\,1}^\infty\left(a_na_{n+1}\right)^{\fra  c12}\ \leqslant\ \sum_{n\,=\,1}^\infty\left(\frac{a_n+a_{n+1}}2\rig  ht)=\frac{a_1}2+\sum_{n\,=\,2}^\infty a_n

    Does this help?
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