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Math Help - complex nos analysis

  1. #1
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    complex nos analysis

    can someone help me with this question pls think i undersdtand trhe concepts but cant quite get the answer thanks

    Edgar

    Let an be a sequence of complex numbers such that sigma|an| converges and
    sigma (an/k^n) = 0 for all k ∈ N. Show that an = 0 for all n.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by edgar davids View Post
    can someone help me with this question pls think i undersdtand trhe concepts but cant quite get the answer thanks

    Edgar

    Let an be a sequence of complex numbers such that sigma|an| converges and
    sigma (an/k^n) = 0 for all k ∈ N. Show that an = 0 for all n.
    Put g(k)=\sum a_n/k^n, then if z=1/k:

    <br />
f(z)=g(1/z)=\sum a_n z^n\equiv 0<br />

    is absolutly convergent inside the unit disk so f is analytic there, and:

    <br />
D^{m}f(0)=m!\ a_m<br />

    but:

    <br />
D^{m}f(0)=0<br />

    so a_m=0 for every m=0, 1, ..

    RonL
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