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Thread: A proof of convergence of an infinite series

  1. #1
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    A proof of convergence of an infinite series

    Show that if $\displaystyle \sum$ a_k is absolutely convergent and abs(b_k) $\displaystyle \leq$ abs(a_k) fopr all k, then $\displaystyle \sum$ b_k is absolutely convergent

    Here is what I have for this:
    Because abs(b_k) $\displaystyle \leq$ abs(a_k), then $\displaystyle \sum$ abs(b_k) $\displaystyle \leq$ $\displaystyle \sum$ abs(a_k)
    With the basic comparison test, since $\displaystyle \sum$ abs(a_k) is convergent, then $\displaystyle \sum$ abs(b_k) is convergent
    Since $\displaystyle \sum$ abs(b_k) is convergent, by theorem $\displaystyle \sum$ bk converges and is absolutely convergent.

    Is this proof sufficient?
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  2. #2
    MHF Contributor

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    Yes that is a proof.
    You may want to add that $\displaystyle {\color{blue}{0 \leqslant}} \sum {\left| {b_n } \right|} $, just to be complete.
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