Question

Use your e result above to find a power series representation of f(x)= x/(2-x)²

Answer 1

`\[f(x) = \sum_(n=0)^(\infty) nx^n.\]`

The power series representation of
`\(f(x) = (x)/((2-x)^2)\) is given by the sum \(\sum_(n=0)^(\infty) nx^n\), where each term is the product of the coefficient \(n\) and a power of \(x\).`

Step-by-step explanation:

The given function
`\(f(x) = (x)/((2-x)^2)\) can be expressed as a power series by first representing \((1)/((1 - u)^(k+1))\) as the sum \(\sum_(n=0)^(\infty) \binom{n+k}{k} u^n\), where \(|u| < 1\). In this case, \(u = (x)/(2)\) and \(k = 1\), giving \((1)/((1 - (x)/(2))^2) = \sum_(n=0)^(\infty) \binom{n+1}{1} \left((x)/(2)\right)^n\).`

Multiplying both sides by
`\(x\), we obtain \(f(x) = \sum_(n=0)^(\infty) n \left((x)/(2)\right)^n\). This is a power series representation of \(f(x)\) in terms of its coefficients \(n\), each multiplied by a power of \((x)/(2)\).`

The resulting series is a representation of
`\(f(x)\) as an infinite sum of terms, each corresponding to the coefficient \(n\) in the expansion. This form provides a concise way to express \(f(x)\)` as an analytical function, particularly useful for various mathematical analyses and computations.