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Complex Radius of Convergence (without limsup formula)

  1. Feb 16, 2012 #1

    Suppose we have that Ʃanzn converges in a radius R, that is, converges for all |z| < R.

    I want to determine the radius of convergence (in terms of R) of some other series WITHOUT relying on the 1/R = limsup{(|an|1/n} formula.

    Namely, the radius of convergence of...

    a) Ʃn3anzn

    b) Ʃanz3n

    c) Ʃan3zn

    In each case, I think it is about relating back to our first condition, that for Ʃanzn we require |z| < R for convergence?
    For example, in b), if we originally required |z| < R, we now require |z3| = |z|3 < R, implying |z| < R1/3, and thus b) having a radius of convergence of R1/3?

    Applying a similar argument to a), we obtain |z| < R/(n3). How can this be interpreted? And in c), we obtain |z| < R/(an2). Again, this doesn't make alot of sense to me. Perhaps this is the wrong approach?

    Any help or hints much appreciated!
  2. jcsd
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