(Solved): as soon as possible please  Find the Maclaurin series for \( f(x) \) using the definition of a ...

Find the Maclaurin series for \( f(x) \) using the definition of a Maclaurin series. [Assume that \( f \) has a power series expansion. Do not show that \( R_{n}(x) \rightarrow 0 . \). Find the associated radius of convergence \( R \). \[ f(x)=2(1-x)^{-2} \] Step 1 The Maclaurin series formula is \( f(0)+f^{\prime \prime}(0) x+\frac{f^{\prime \prime}(0)}{2 !} x^{2}+\frac{f^{\prime \prime \prime}(0)}{3 !} x^{3}+\frac{f^{(4)}(0)}{4 !} x^{4}+\cdots{ }^{4} \). \( f(x)=2(1-x)^{-2} \) has the following derivatives. \[ \begin{array}{ll} f^{\prime}(x)= & (1-x) \\ f^{\prime \prime}(x)= & (1-x) \\ f^{\prime \prime \prime}(x)= & (1-x) \\ f^{(4)}(x)= & (1-x) \end{array} \]