In the hydrogen spectrum, the Lyman series limit arises from the \( n = 2 \) to \( n = \infty \) transition, and the Paschen series limit from the \( n = 4 \) to \( n = \infty \) transition. The Rydberg formula, \( \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \), relates wavelength \( \lambda \) to energy levels \( n_1 \) and \( n_2 \), with \( R_H \) as the Rydberg constant. For the Lyman series limit (\( n = 2 \) to \( n = \infty \)): \[ \frac{1}{\lambda_L} = R_H \left( \frac{1}{2^2} - \frac{1}{\infty^2} \right) = R_H \left( \frac{1}{4} \right) \] For the Paschen series limit (\( n = 4 \) to \( n = \infty \)): \[ \frac{1}{\lambda_P} = R_H \left( \frac{1}{4^2} - \frac{1}{\infty^2} \right) = R_H \left( \frac{1}{16} \right) \] The ratio of wavelengths is: \[ \frac{\lambda_L}{\lambda_P} = \frac{16}{4} = 4 \] Therefore, the ratio of the wavelengths of the Lyman series limit to the Paschen series limit is \( \frac{9}{9} \), simplifying to \( 1 \).