Final answer:
To find a power series for the function f(x) = 1/(1-x)², we can differentiate the power series representation of 1/(1-x) twice. The resulting power series is Σ n*(n-1)*xⁿ⁻² (n=0 to infinity).
Step-by-step explanation:
To find a power series for the function f(x) = 1/(1-x)², we can start with the power series representation of 1/(1-x) which is Σ xⁿ (n=0 to infinity). To obtain the power series for f(x), we can differentiate the power series representation of 1/(1-x) twice. Differentiating the series twice will result in a new series with coefficients that are related to the coefficients of the original series.
Let's start by differentiating the power series Σ xⁿ with respect to x. Each term xⁿ will become n*xⁿ⁻¹. So the new series becomes Σ n*xⁿ⁻¹ (n=0 to infinity).
Now, let's differentiate the new series Σ n*xⁿ⁻¹ with respect to x. Each term n*xⁿ⁻¹ will become n*(n-1)*xⁿ⁻². So the resulting series becomes Σ n*(n-1)*xⁿ⁻² (n=0 to infinity).
Therefore, the power series for f(x) = 1/(1-x)² is Σ n*(n-1)*xⁿ⁻² (n=0 to infinity).