The power series representation for the function x³/(1-x³) is obtained by recognizing it as a geometrical series with a common ratio x^3 and then applying the geometric series summation formula.
The function given is x³/(1-x³). To find the power series representation for the function, we recognize that the denominator resembles a geometric series. A geometric series has the form 1/(1-r), where r is the common ratio. In this case, our r is x³ The power series for a geometric series is given by sum_{n=0}^{infinity} r^n. Thus, we can write the power series representation of the function as sum_{n=0}^{infinity} x^{3n+3}. We achieve this by substituting x^3 as the r into the geometric series and multiplying by x^3 to get the original function as required.