Answer
$$\frac{\Delta \lambda}{\lambda_{\text {rest }}}=\frac{v}{c}$$
Work Step by Step
$\text{to solve this problem we can use Eq. $33-3$ states }$
$$\lambda=\lambda_{\text {rest }} \sqrt{\frac{1+v / c}{1-v / c}}$$
$\text{therefore by using the binomial expansion }$
$$\lambda=\lambda_{\text {rest }} \sqrt{\frac{1+v / c}{1-v / c}}=\lambda_{\text {rest }}\left(1+\frac{v}{c}\right)^{1 / 2}\left(1-\frac{v}{c}\right)^{1 / 2} $$$$\approx \lambda_{\text {rest }}\left(1+\frac{1}{2} \frac{v}{c}\right)\left(1-\left(-\frac{1}{2}\right) \frac{v}{c}\right)=\lambda_{\text {rest }}\left(1+\frac{1}{2} \frac{v}{c}\right)^{2}$$
$$\lambda=\lambda_{\text {rest }}\left(1+2\left(\frac{1}{2} \frac{v}{c}\right)\right)=\lambda_{\text {rest }}\left(1+\frac{v}{c}\right) $$$$=\lambda_{\text {rest }}+\lambda_{\text {rest }} \frac{v}{c} \rightarrow \lambda-\lambda_{\text {rest }}=\Delta \lambda=\lambda_{\text {rest }} \frac{v}{c} \rightarrow $$
$$\frac{\Delta \lambda}{\lambda_{\text {rest }}}=\frac{v}{c}$$
