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Thread: Complex Series

  1. #1
    Member mohammadfawaz's Avatar
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    Complex Series

    Hello!
    Suppose that $\displaystyle \{z_n\}$ is a sequence of complex numbers satisfying $\displaystyle Re\{z_n\}\geq0$ and that the two series $\displaystyle \sum_{n=1}^\infty z_n$ and $\displaystyle \sum_{n=1}^\infty z_n^2$ are convergent. Prove that the series $\displaystyle \sum_{n=1}^\infty |z_n|^2$ is also convergent.

    Well, assuming $\displaystyle z_n=x_n+iy_n$, we can deduce from the given that the following series converge:
    $\displaystyle \sum_{n=1}^\infty x_n$, $\displaystyle \sum_{n=1}^\infty |x_n|$,
    $\displaystyle \sum_{n=1}^\infty y_n$,
    $\displaystyle \sum_{n=1}^\infty x_n^2-y_n^2$,
    $\displaystyle \sum_{n=1}^\infty 2x_n y_n$.

    The sequence $\displaystyle \sum_{n=1}^\infty |z_n|^2$ can be written as:
    $\displaystyle \sum_{n=1}^\infty x_n^2+y_n^2$ or $\displaystyle \sum_{n=1}^\infty |x_n^2-y_n^2+2ix_ny_n|$

    I am stuck here for now. Any ideas?

    Thanx

    Mohammad
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Quote Originally Posted by mohammadfawaz View Post
    Suppose that $\displaystyle \{z_n\}$ is a sequence of complex numbers satisfying $\displaystyle Re\{z_n\}\geq0$ and that the two series $\displaystyle \sum_{n=1}^\infty z_n$ and $\displaystyle \sum_{n=1}^\infty z_n^2$ are convergent. Prove that the series $\displaystyle \sum_{n=1}^\infty |z_n|^2$ is also convergent.
    If the series of positive terms $\displaystyle \sum x_n$ converges, then so does $\displaystyle \sum x_n^2$ (comparison test, because $\displaystyle x_n<1$ for all sufficiently large n). If you also know that $\displaystyle \sum(x_n^2-y_n^2)$ converges, then (by subtraction) so does $\displaystyle \sum y_n^2$. Hence so does $\displaystyle \sum(x_n^2+y_n^2).$
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