Show that the first few frequencies of light that is emitted when electrons fall to nth level from levels higher than n, are approximate harmonics (i.e., in the ratio 1: 2: 3...) when n>>1.

Hint: Use the formula of frequency for a particular transition.
Step 1: Find the frequency of any line series for a particular transition.
The frequency of any line in a series in the spectrum of hydrogen-like atoms corresponding to the transition of electrons from (n+p) level to nth level can be expressed as a difference of two terms:
νmn=cRZ21(n+p)2-1n2
where, m=n+p, (p=1,2,3,...) and R is Rydberg constant.
For p<<nνmn=cRZ2[1n2(1+pn)-2-1n2]νmn=cRZ2[1n2-2pn3-1n2][By binomial theorem (1+x)n=1+nx if x<1]νmn=cRZ22pn3=(2cRZ2n3)p
Thus, the first few frequencies of light that is emitted when electrons fall to the nth level from levels higher than n, are approximate harmonic (i.e., in the ratio 1 : 2 : 3 ...) when n>>1.