When an individual gaseous element such as hydrogen is excited (by heating or an electrical discharge) and its emitted light is examined through a spectrometer, the result is not a smooth, continuous rainbow of colours but rather a set of sharp, discrete bright lines at very specific wavelengths - a line spectrum, unique to that particular element, functioning almost like a fingerprint that lets scientists identify a gas's composition from its light alone (including the composition of distant stars and the Sun).
For hydrogen specifically, every observed spectral-line wavelength is given by a single Rydberg-type formula,
λ1=R(n21−m21),m>n positive integers
where R=1.09737×107 m−1 is the Rydberg constant. Fixing the lower level n and letting m range over every larger integer generates one complete named series of spectral lines: n=1 gives the Lyman series (ultraviolet region); n=2 gives the Balmer series (visible region - the only series bright enough to be seen directly with the eye); n=3 gives the Paschen series (near infrared); n=4 gives the Brackett series (mid infrared); and n=5 gives the Pfund series (far infrared). Within any one series, successive lines (as m increases) crowd progressively closer together, approaching a limiting wavelength - the series limit - as m→∞.
This same formula also lets the total number of distinct spectral lines an excited hydrogen sample can emit be counted: if an electron has been excited up to level n, it can subsequently make a transition down to any lower level, and the total number of distinct transitions (and hence distinct emitted wavelengths) possible works out to n(n−1)/2. …