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Usually I see power series such as $$\sum_{n=0}^\infty \frac{n!}{(n)^n}z^n$$ Now, I am asked to find the radius of convergence for $$\sum_{n=0}^\infty 2^{-n}z^{n^2} \quad \text{ and } \quad \sum_{n=1}^\infty (3+4i)^n(z-4i)^n$$

How would I find the radius of convergence for those two power series?

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    What do you expect them to be? For example, you know that if $z = 1$ in the first one, it (---). If $z >1$, then it (---). The second is funny-looking, but I suspect that you suspect that if z is near $4i$ then it converges. Note also that $|3 + 4i| = 25$.2011-11-10
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    @mixedmath Actually $|3+4i|=5$.2011-11-10
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    One general way: make an educated guess, then try to prove said guess. Also note you should center the second series around $z=4i$, so maybe write $w=z-4i$. @mixedmath: 52011-11-10
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    @Hardy: oh ... you're right... that's embarassing. And it's the most common right triangle. My pride hurts, so I'm going to go hide for a while.2011-11-10

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