Question

Use the power series

[infinity]
1/1-x = Σ xⁿ , |x| < x
n=0

to find a power series for the function, centered at 0.
f(x) = 1/(1-x)²

[infinity]
f(x)=Σ =
n=1

Answer 1

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).