Part 4

1. Find the interval of convergence (you must test endpoints, if applicable) of the following series.
   
   $$\sum_{n=0}^{\infty } 2^{-n} \left( 2x-1 \right )^n$$
   
   

2. Find a power series representation, and the interval on which it represents the function, for
   
   $$f(x)= \frac{x^2}{4-x^2}$$
   
   

3. Find the Taylor series centered at $a=8$ for
   
   $$f(x)=\sqrt[3]{x}$$
   
   

4. Using Taylor's Inequality, determine an upper bound for the error in approximating $f(x)=e^{-2x}$ by its Taylor polynomial $T_2(x)$, centered at $a=0$, on the interval $[-1/4,1/4]$.