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Math Help - Convergence of Infinite Series

  1. #1
    Member mohammadfawaz's Avatar
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    Convergence of Infinite Series

    Hello,

    Suppose 0<\alpha<\frac{\pi}{2} and \{z_n\} is a sequence of complex numbers satisfying -\alpha<arg(z_n)<\alpha. Prove that the two infinite series \sum_{n=1}^\infty {z_n} and \sum_{n=1}^\infty |{z_n}| are both convergent or both divergent.

    First, I'm trying to show that if the first converges, then the second also converges. Setting z_n=x_n+iy_n, I know that all x_n are positive and hence \sum_{n=1}^\infty |x_n| converges. But for y_n, this is still unclear for me.

    Thank you

    Mohammad
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    For all n we have x_n\neq 0 (why?). Then,

    \left |\dfrac{y_n}{x_n}\right |=\left | \tan (\arg z_n)\right |\leq \left |{\tan \alpha}\right |=r<+\infty

    So, |y_n|\leq r|x_n| for all n and \sum_{n=1}^{+\infty}r|x_n| is convergent which implies \sum_{n=1}^{+\infty}|y_n| is convergent.
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