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Math Help - absolutely convergent series

  1. #1
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    absolutely convergent series

    Show that if a1+a2+a3+... is an absolutely convergent series of real numbers, then a1^2+a2^2+a3^2+... converges.
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    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Milus View Post
    Show that if a1+a2+a3+... is an absolutely convergent series of real numbers, then a1^2+a2^2+a3^2+... converges.
    Consider the Cauchy-Schwarz inequality.
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    Quote Originally Posted by Mathstud28 View Post
    Consider the Cauchy-Schwarz inequality.

    Can you help me more.I really struggle with these proofs.The concept seems clear but I have no idea how to show it.
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    Quote Originally Posted by Milus View Post
    Show that if a1+a2+a3+... is an absolutely convergent series of real numbers, then a1^2+a2^2+a3^2+... converges.
    Hint: For all sufficiently large n, |a_n| must be less than 1, and therefore a_n^2<|a_n|.
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    Quote Originally Posted by Opalg View Post
    Hint: For all sufficiently large n, |a_n| must be less than 1, and therefore a_n^2<|a_n|.
    Can you help me more.I just cannot connect it to one piece.
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    Quote Originally Posted by Milus View Post
    Can you help me more.I just cannot connect it to one piece.
    This is true: \left| x \right| \leqslant 1 \Rightarrow \quad x^2  \leqslant x.
    Because \left( {a_n } \right) \to 0 \Rightarrow \quad \left( {\exists N} \right)\left[ {n \geqslant N \Rightarrow \left| {a_n } \right| < 1 \Rightarrow \quad \left( {a_n } \right)^2  \leqslant \left| {a_n } \right|} \right].

    Now by direct comparison \sum\limits_n {\left( {a_n } \right)^2 } must converge.
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    Is that sufficient proof or shoule there be added something????
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    Quote Originally Posted by Milus View Post
    Is that sufficient proof or shoule there be added something????
    Since for n\geq N (sufficiently large n) we have that a_n^2 \leq |a_n| it follows that \sum_{n=N}^{\infty} a_n^2 \leq \sum_{n=N}^{\infty} |a_n| < \infty since \{ a_n\} is absolutely convergent.
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