Consider $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = {x}^{1/3}$. let g(x) = $\sum_{n=0}^{∞} a_n(x − 3/2)^n$ where $a_n= \frac {{f}^{(n)}(3/2)}{n!}$ for $n≥0$. what is the largest open set contained in {${x|f(x)=g(x)} $}?
my answer : i think largest open set will be $(-3/2,0) \cup (0,3/2)$ ...
is its correct or not ? pliz tell me
or give me any hints or if u have time pliz tell me the solution i would be more thankful..
Complex analysis fact: power series converge up to the nearest singularity of the function. In the case of your series, we have a singularity at $x = 0$ and the radius of convergence will be $3/2$...
Alternatively, you can calculate explicitly the coefficients of the series, check the radius of convergence and prove that converges to $f$ applying Taylor.