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Math Help - what is the radius of convergence of that power series

  1. #1
    Junior Member sorv1986's Avatar
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    Cool what is the radius of convergence of that power series

    Consider the function f(z) = 1 / (1+z) where z ∈ C and let f(z) = Σ an (z-a)ⁿ be the

    Taylor series expansion of f(z) about the point a ∈ R.

    then what is the radius of convergence of the power series?

    thanx in advance. regards
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: what is the radius of convergence of that power series

    Deleted (I misread the question)
    Last edited by FernandoRevilla; November 24th 2011 at 01:30 AM.
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  3. #3
    Junior Member sorv1986's Avatar
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    Re: what is the radius of convergence of that power series

    Quote Originally Posted by FernandoRevilla View Post
    Hint Use the substitution z=w-a and expand f(z) by means of a geometric series .
    thnx for ur kind reply.
    But how to determine the radius of convergence?

    as you said,

    f(w-a)=1/(1+(w-a))=(1+C)⁻=1-(w-a)+(w-a)⁴-...........=Σ(-1)ⁿ (w-a)ⁿ [the expansion is valid if (w-a)<1]

    Thean what should i do? confused a bit.


    thanxx
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Re: what is the radius of convergence of that power series

    Quote Originally Posted by sorv1986 View Post
    Thean what should i do? confused a bit.
    Sorry, I misread the question, f is analytic in \mathbb{C}-\{i,-i\} . By a well known theorem the radius of convergence is the distance of a to the nearest singularity, that is R=\sqrt{a^2+1} .
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