Use the geometric series formula 11y=n=0yn\displaystyle \frac{{1}}{{{1}-{y}}}={\sum_{{{n}={0}}}^{{\infty}}}{y}^{{n}} to express the function as a series:
x291x57\displaystyle \frac{{{x}^{{\frac{{2}}{{9}}}}}}{{{1}-{x}^{{\frac{{5}}{{7}}}}}} =n=0\displaystyle ={\sum_{{{n}={0}}}^{{\infty}}}  \displaystyle