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The Lyman and Balmer series refer to wavelengths that quantify a transition to a specific energy level. The Lyman series refers to transitions of an electron from one energy level to the n=1 energy level. The Balmer series refers refers to transitions of an electron from one energy level to the n=2 energy level.
To clarify a little bit more on the Balmer series, the Balmer series of spectral emissions lines of hydrogen is the electron transitions going from a higher energy level back down to the principal quantum number of 2. To better visualize or think about this n1 (principal quantum number)=2 and n2=3,4,5... in the Balmer series whereas in the Lyman series n1=1. The Balmer series is also situated in the visible light region of the spectrum while the Lyman series is situated in the UV region of the spectrum. If you want to read more about the Balmer and Lyman series and better understand the differences between the two I recommend you read 1.A3 Atomic Spectra page 7.
The Lynman and Balmer series can be seen on the light emission spectrum; the Lynman series refers to when electrons fall to energy level n=1, and the Balmer series refers to when electrons fall to energy level n=2. You can see this on the chart he provided on the slide by seeing which line the arrow goes to: if it goes to the first/bottom line, it is within the Lynman series, and if it falls to the second line, it is within the Balmer series.
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