(Solved): 1. Test the convergence of the power series below: 10 \[ \sum_{n=1}^{\infty} \frac{(1-2 n)^{3}}{(- ...

1. Test the convergence of the power series below: 10 \[ \sum_{n=1}^{\infty} \frac{(1-2 n)^{3}}{(-4)^{n}}(x-3)^{n} \] Calculate the following: a. Center of convergence b. Radius of convergence c. interval of convergence 2. Express as a single power series form: \[ \sum_{n=0}^{\infty}(n-2) a_{n+1}(2 x-3)^{n+1}+\sum_{n=2}^{\infty} a_{n-1}(2 x-3)^{n-1} \] 3. For the function \( y=x \ln (x+1) \), determine its a. MacLaurin Series expansion up to fifth degree. b. Taylor Series expansion up to fifth degree at \( x_{0}=2 \). 4. Test the point and find the solution to the differential equation below: \[ y^{\prime \prime}+x^{2} y^{\prime}+x y=0 \text { at } x_{0}=0 \] 5. Find the solution to the differential equations below: a. \( 3 x^{2} y^{\prime \prime}+6 x y^{\prime}+y=0 \) b. \( x^{2} y^{\prime \prime}+3 x y^{\prime}=0 \), given \( y(1)=0, y^{\prime}(1)=4 \)