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The Balmer series, for example, is formed when the electron jumps from $n_2 = 3, 4, 5, \ldots, ∞$ to $n_1 = 2.$

The Humphry Davy series (The last series is formed when electron jumps from $n_2 = 7, 8, 9, \ldots, ∞$ to $n_1 = 6.$

Can't there be another series formed when electron jumps from $n_2 = 8,9,10... \ldots, ∞$ to $n_1 = 7?$

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    $\begingroup$ There can be and surely is another series, and another after that, and another after that, and another after that. It is just that we ran out of names. After all, infinity is infinite, and the number of guys who ever lived is finite (that is, a bit smaller). $\endgroup$ – Ivan Neretin Jun 26 at 4:51
  • $\begingroup$ I agree with Ivan. There are certainly other series that are theoretically known but are yet practically observed by scientists. Hence there are no name for the series yet. $\endgroup$ – Nilay Ghosh Jun 26 at 7:59
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    $\begingroup$ @NilayGhosh They are observed, it's rather a matter of notability. $\endgroup$ – Mithoron Jun 26 at 16:03
  • $\begingroup$ Note also that for a large enough value of $n$. The differences between energy level $n$ and $n+1$ is small enough that they are very similar. As an exercise, you can try to figure out the critical value of $N$ such that for $n>N$, the difference in energy between levels $n$ and $n+1$ is less than threshold $E$. That's just a long way to say, as $n$ increases, the differences between series get small. $\endgroup$ – Zhe Jun 27 at 21:16
  • $\begingroup$ @Zhe 1.What does threshold E mean though? 2. I understood the part where you said that the energy difference gets smaller as the radius increases.. Does that mean that if I have a REALLY SMALL energy difference it won't form a spectral line, or is there some other explanation? $\endgroup$ – Happy Unicorn Jun 30 at 9:16
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As Ivan commented, there are actually an infinite number of possible series of this type. So your question is really why there are only six named series. The reason is part of the culture of science. Typically, a result is named if it is, to use Mithoron's term, sufficiently notable. Thus we have the Diels-Alder reaction, the Hartree–Fock method, the Wentzel–Kramers–Brillouin (WKB) approximation, the Southern blot, and so on.

Hence the reason why the initial series of hydrogen spectra were named is because they were notable scientific results. [In addition—and perhaps someone more expert in the history of science can comment on this—it is possible that identifying the, say, fifth and sixth series weren't that scientifically notable, but a tradition had become established of naming successive spectra of hydrogen.] But beyond the sixth, finding more series was essentially deemed routine. The last published series I know of is the seventh, identified by Hansen and Strong, but not named after them [Peter Hansen and John Strong, "Seventh Series of Atomic Hydrogen," Appl. Opt. 12, 429-430 (1973)].

That's not to say it wouldn't be experimentally difficult to see higher-numbered series, since they become increasingly faint. Rather, it's that, at this point, finding them isn't interesting science.

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