The function $f(x)=\frac{1}{1-x}$ has the Maclaurin series shown below with interval of convergence $).$

${\displaystyle \underset{n=0}{\overset{}{}}}{x}^n$

Complete each of the following:

a$)$ Write the Maclaurin polynomial order $3$ for $f(x).(4$ points$)$

b$)$ Write the Maclaurin series for $g(x)=\frac{1}{1+2x}$ in summation notation. $(4$ points$)$

c$)$ What is the interval of convergence for $g(x)?(4$ points$)$

d$)$ Use your answer to part $b$ to determine the Maclaurin series for $g^{\text{'}}(x)(4$ points$).$

e$)$ What function is represented by the series you found in part d$?(4$ points$)$