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Anyone knows anything about the Radius on convergence in the complex plane (Complex Analysis)

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- Thread starter racland
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Anyone knows anything about the Radius on convergence in the complex plane (Complex Analysis)

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Hope I was helpful

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ranger

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Hope I was helpful

:rofl: :rofl:

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HallsofIvy

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The problem states:

Find the Radius of Convergence of the following Power Series:

(a) Sumation as n goes from zero to infinity of Z^n!

(b) Sumation as N goes from zero to infinity of (n + 2^n)Z^n

For (a) I think the radius of convergence is 1 but I'm a bit unsure of that...

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HallsofIvy

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[tex]\lim_{n\rightarrow \infty} \left|\frac{a_{n+1}}{a_n}\right|[/tex]

is less than 1.

(Diverges if that limit is greater than one, may converge absolutely or conditionally or diverge if it is equal to 1).

In particular, for Z

Try (n+ 2

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StatusX

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Is that [itex]z^{n!}[/itex]? If so, you can prove a general result that if a_n is any increasing sequence of natural numbers, then [itex]z^{a_n}[/itex] converges iff |z|<1. This is the case HallsofIvy did if a_n=n, and yours if a_n=n!. The general proof follows from the result for a_n=n (which is the smallest increasing sequence of natural numbers).

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