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Math Help - Real Analysis - Series

  1. #1
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    Real Analysis - Series

    a. Show that if Σan converges absolutely, then Σ(an)^2 also converges absolutely. Does this proposition hold without absolute convergence?

    b. If Σan converges and an >= 0, can we conclude anything about Σsqrt(an)?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by ajj86 View Post
    a. Show that if Σan converges absolutely, then Σ(an)^2 also converges absolutely. Does this proposition hold without absolute convergence?
    As \sum a_n converges eventualy a_n<1, then a_n^2<a_n and you get the partial sums of \sum a_n^2 eventualy form an increasing bounded sequence and hence the series converges.

    If you only have conditional convergence this does not hold, a counter example will suffice to prove this: put a_n=(-1)^n n^{-1/2} this will give a conditionaly convergent series, but a_n^2=n^{-1} and the corresponding series will diverge as its the harmonic series.

    CB
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    Grand Panjandrum
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    Quote Originally Posted by ajj86 View Post
    b. If Σan converges and an >= 0, can we conclude anything about Σsqrt(an)?
    consider a_n=1/n^2, then consider a_n=1/n^4

    CB
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