(Solved): Use the formula for the binomial series: \[ \begin{array}{l} \qquad(1+x)^{m}=1+m x+\frac{m(m-1)}{2 ...

Use the formula for the binomial series: \[ \begin{array}{l} \qquad(1+x)^{m}=1+m x+\frac{m(m-1)}{2 !} x^{2}+\cdots+\frac{m(m-1) \cdots(m-k+1)}{k !} x^{k}+\cdots \\ \text { to obtain the Maclaurin series for } \frac{1}{(1+x)^{3}} \text {. } \\ 1-3 x+\frac{1}{2 !} \sum_{k=2}^{\infty}(-1)^{k} \frac{(k+4) !}{k !} x^{k} \frac{m(m-1) \cdots(m-k+1)}{k !} x^{k} \text { if }|x|<1 \\ 1-3 x+\frac{1}{2 !} \sum_{k=2}^{\infty}(-1)^{k} \frac{(k+2) !}{k !} x^{k} \\ 1-3 x+\frac{1}{4 !} \sum_{k=2}^{\infty}(-1)^{k+1} \frac{(k+2) !}{k !} x^{k} \\ 1+\frac{1}{3 !} \sum_{k=1}^{\infty} \frac{(k+3) !}{k !} x^{k} \\ 1+3 x+\sum_{k=2}^{\infty} \frac{(k+2) !}{k !} x^{k} \end{array} \]