Loser66
  • Loser66
Determine the radius of convergence of \(\sum_{n=0}^\infty \left(\begin {matrix}a\\n\end{matrix}\right)(z-1)^n\) Please, help
Mathematics
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SOLVED
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katieb
  • katieb
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Loser66
  • Loser66
assuming a is not a positive integer
Loser66
  • Loser66
I am not allowed to use ratio or root test.
Kainui
  • Kainui
Oh it might be easier if you use \binom{a}{n} to make \[\binom{a}{n}\] But as far as this question goes... I don't know. Possibly some thing about circle centered at z=1 beats me, I need to learn complex analysis.

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anonymous
  • anonymous
Do you know about Dirichlet's test?
anonymous
  • anonymous
Not that it would be useful, but it'd be nice to know what tests you *can* use.
Loser66
  • Loser66
I think either Dirichlet or Hadamard
anonymous
  • anonymous
The Hadamard test is a lot like the root test, though...
Loser66
  • Loser66
but my prof likes it. He proved it in class but not root test.
anonymous
  • anonymous
Alright then. The Hadamard theorem says that the radius of convergence \(R\) satisfies \[\frac{1}{R}=\limsup_{n\to\infty}|c_n|^{1/n}=\limsup_{n\to\infty}\left|\frac{a!}{n!(a-n)!}\right|^{1/n}\]And \(a\) is *not* a positive integer?
Loser66
  • Loser66
yup
Loser66
  • Loser66
or we can go from f(z ) = z^a and a is not a positive integer
anonymous
  • anonymous
Is that to say it's also not necessarily true that \(a\in\mathbb{R}\)? I'm not sure if the binomial coefficent/Gamma function is equipped to handle complex \(a\).
Loser66
  • Loser66
But the question asked to go from Taylor series of f(z) = z ^ a
Loser66
  • Loser66
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