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Suppose we have that Ʃa_{n}z^{n}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{(|a_{n}|^{1/n}} formula.

Namely, the radius of convergence of...

a) Ʃn^{3}a_{n}z^{n}

b) Ʃa_{n}z^{3n}

c) Ʃa_{n}^{3}z^{n}

In each case, I think it is about relating back to our first condition, that for Ʃa_{n}z^{n}we require |z| < R for convergence?

For example, in b), if we originally required |z| < R, we now require |z^{3}| = |z|^{3}< R, implying |z| < R^{1/3}, and thus b) having a radius of convergence of R^{1/3}?

Applying a similar argument to a), we obtain |z| < R/(n^{3}). How can this be interpreted? And in c), we obtain |z| < R/(a_{n}^{2}). Again, this doesn't make alot of sense to me. Perhaps this is the wrong approach?

Any help or hints much appreciated!

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# Complex Radius of Convergence (without limsup formula)

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