The spectral series in the visible region for a hydrogen atom are the Balmer series. The Balmer series consists of several lines in the visible spectrum, which are produced when electrons transition from higher energy levels to the second energy level (n=2) in a hydrogen atom. These transitions result in the emission of photons with specific energies, corresponding to different wavelengths of light in the visible region.

The Balmer series is characterized by the following transitions:

- The shortest wavelength in the Balmer series is the H-alpha line, which corresponds to the transition from the third energy level (n=3) to the second energy level (n=2). This line appears as a red color.
- The next line is the H-beta line, corresponding to the transition from the fourth energy level (n=4) to the second energy level (n=2). This line appears as a blue-green color.
- The H-gamma line corresponds to the transition from the fifth energy level (n=5) to the second energy level (n=2). This line appears as violet.
- The H-delta line corresponds to the transition from the sixth energy level (n=6) to the second energy level (n=2). This line appears as ultraviolet.

The ratio of the maximum to the minimum wavelengths in the Balmer series can be calculated by considering the transitions between the highest and lowest energy levels involved in the series.

- The maximum wavelength corresponds to the transition from the infinity energy level (n=∞) to the second energy level (n=2). This transition is known as the Lyman limit and is in the ultraviolet region.
- The minimum wavelength corresponds to the transition from the third energy level (n=3) to the second energy level (n=2), which is the H-alpha line in the Balmer series.

Calculating the ratio of the maximum to minimum wavelengths:

1. The energy of the transition from the infinity energy level to the second energy level can be calculated using the Rydberg formula:

1/λ = R (1/2^2 - 1/n^2)

where λ is the wavelength, R is the Rydberg constant, and n is the principal quantum number.

2. Substituting n=∞ and n=2 into the formula, we get:

1/λ_max = R (1/2^2 - 1/∞^2)

Since 1/∞^2 is approximately zero, the term becomes:

1/λ_max ≈ R/4

3. The energy of the transition from the third energy level to the second energy level can be calculated in a similar manner:

1/λ_min = R (1/2^2 - 1/3^2)

4. Taking the ratio of λ_max to λ_min:

(1/λ_max) / (1/λ_min) = (R/4) / (R/4 - R/9)

Simplifying the expression:

(1/λ_max) / (1/λ_min) = 9/5

Therefore, the ratio of the maximum to minimum wavelengths in the Balmer series is 9:5.

In conclusion, the Balmer series of a hydrogen atom in the visible region consists of several lines, including the H-alpha, H-beta, H-gamma, and H-delta lines. The