Question

Determine the radius of convergence of \(\sum_{n=0}^\infty \left(\begin {matrix}a\\n\end{matrix}\right)(z-1)^n\)
Please, help

Answer 1

assuming a is not a positive integer

Answer 2

I am not allowed to use ratio or root test.

Answer 3

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.

Answer 4

Do you know about Dirichlet's test?

Answer 5

Not that it would be useful, but it'd be nice to know what tests you *can* use.

Answer 6

I think either Dirichlet or Hadamard

Answer 7

The Hadamard test is a lot like the root test, though...

Answer 8

but my prof likes it. He proved it in class but not root test.

Answer 9

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?

Answer 10

yup

Answer 11

or we can go from f(z ) = z^a and a is not a positive integer

Answer 12

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 \(a\).

Answer 13

But the question asked to go from Taylor series of f(z) = z ^ a