Explanation

(a) For maximum wavelength of Balmer series
$\frac{1}{\lambda_{\max}} = R\left( \frac{1}{2^{2}} - \frac{1}{3^{2}} \right) = \frac{R\ \times \ 5}{36}$ …(i)
For minimum wavelength of Balmer series,
$\frac{1}{\lambda_{\min}} = R\left( \frac{1}{2^{2}} - \frac{1}{\ \infty} \right) = \frac{R\ }{4}$ …(ii)
From Eqs.(i)and (ii), we have
$\therefore\ \ \ \ \ \ \ \ \ \ \ \frac{\lambda_{\min}}{\lambda_{\max}} = \frac{R\ \times \ 5}{36} \times \frac{4\ }{R} = \frac{5\ }{9}$