Question

Represent the function f(x) = $$x^{0.5}$$ as a power series.


Find the interval of convergence.

Answer 1

I'm looking for the interval of convergence.

Answer 2

according to wikipedia, fractional powers are not allowed in a power series
http://en.wikipedia.org/wiki/Power_series#Examples

Answer 3

Ah, interesting. How do I find the interval of convergence of the Puiseux series of $$x^{0.5}$$?

Answer 4

According to this website, you can make a taylor series, but you should center it at a=1 or some other positive number, since the derivatives do not exist at a = 0.

http://math.stackexchange.com/questions/597731/compute-the-first-five-non-zero-terms-of-the-taylor-series-about-a-4-for-fx?rq=1

Answer 5

in the example they centered it at a = 4

Answer 6

the only problem is, i dont see a nice closed form for the power series.
here is another attempt at it, where they centered it about a = 1
http://math.stackexchange.com/questions/442563/taylor-series-expansion-for-fx-sqrtx-for-a-1

here it looks like they were able to get a closed form

Answer 7

so it depends on what you want to do .

Answer 8

im sorry , if it was centered at 3 what would be the interval of convergence?

Answer 9

\(x^{1/2}=\left((x-3)+3\right)^{1/2}\). Then use binomial series.

Answer 10

oh thank you :)