Finding the radius of convergence from a closed form representation of a series

Asked Mar 22 '14 at 21:22

Active Mar 22 '14 at 21:22

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Say we have the following:

$$\sum_{n=0}^\infty a_n = \frac{\sqrt{5-x}+45\sqrt{x}}{(5-x)^{\frac{a}{b}}} $$ I have made this up on the spot and by no means is this true. How would I go by finding the radius of convergence by the closed form? Do I just pick values of x in which the absolute value would not yield negative numbers inside the roots? This might be an elementary question but I am rusty.

asked Mar 22 '14 at 21:22

H5159

It depends on whether you want the sum to converge to a real or complex number – Mar 22 '14 at 22:14
I'm assuming real number. – H5159 Mar 22 '14 at 22:45
Then you choose $x>5$. However, I would like to ask - what is $x$? – Mar 22 '14 at 22:48
You mean $x<5$ correct? And don't we go with the absolute value? – H5159 Mar 22 '14 at 22:59
Yes, I meant $|x|<5$ - with the absolute value. – Mar 22 '14 at 23:01
If we were to allow complex numbers, how would this change the convergence? – H5159 Mar 22 '14 at 23:40
It would converge for all $x$, with the condition that $x\neq 5,b\neq0$. – Mar 22 '14 at 23:41

Finding the radius of convergence from a closed form representation of a series

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