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Math Help - Sequences and sums

  1. #1
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    Sequences and sums

    Suppose that (x_k)_{k=1}^{\infty} and (y_k)_{k=1}^{\infty} are sequences of positive real numbers, and that \sum_{k=1}^{\infty}x_{k}=S and \sum_{k=1}^{\infty}y_{k}=T. Let z_k=\max(x_k,y_k). Prove that the sum \sum_{k=1}^{\infty}z_k exists.

    Let S_n=\sum_{k=1}^{\infty}x_{k} and let
    Let T_n=\sum_{k=1}^{\infty}y_{k}

    We know S_n coverges to S and T_n converges to T. So S_n+T_n converges to S+T. Am i on the right track? I can't go any further from here so any help would be greatly appreciated.

    Thanks
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  2. #2
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    Quote Originally Posted by charikaar View Post
    Suppose that (x_k)_{k=1}^{\infty} and (y_k)_{k=1}^{\infty} are sequences of positive real numbers, and that \sum_{k=1}^{\infty}x_{k}=S and \sum_{k=1}^{\infty}y_{k}=T. Let z_k=\max(x_k,y_k). Prove that the sum \sum_{k=1}^{\infty}z_k exists.

    Let S_n=\sum_{k=1}^{\infty}x_{k} and let
    Let T_n=\sum_{k=1}^{\infty}y_{k}

    We know S_n coverges to S and T_n converges to T. So S_n+T_n converges to S+T. Am i on the right track? I can't go any further from here so any help would be greatly appreciated.

    Thanks
    You might be able to make use of the fact that z_k < x_k + y_k.
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