During my study of a textbook on mathematical analysis, I encountered a problem that involves estimating the coefficients $a_n$ of the power series expansion of the function:
$$ S(x)=\frac{2}{3}(1+x)^{-2/3}(1-x)^{-4/3}. $$
The textbook provides the following hints to solve the problem:
- Firstly, obtain the expression for the coefficients $a_n$ using the Cauchy product, and then utilize the Euler integral to calculate $$ a_n=\frac{n}{\Gamma(2/3)\Gamma(1/3)}\int_{-1}^1\frac{\sqrt[3]{1+s}}{\sqrt[3]{1-s}}s^{n-1}ds; $$
- Based on the expression above, further deduce that $$ a_n\sim \frac{\sqrt[3]{2n}}{\Gamma(1/3)}, \quad \text{as } n\rightarrow +\infty, $$ where $\Gamma(s)=\int_0^\infty x^{s-1}e^{-x}dx$ denotes the usual Gamma function.
I’m uncertain about how to proceed with these two steps in order to obtain the desired result. Any assistance or guidance would be greatly appreciated.