How to develop in series as in the image below? What does the third term signify?
It is a simple Taylor expansion of the relativistic energy.
What does the third term signify?
It doesn't have a name or any particular meaning as far as I am aware.
If you are measuring the energy of particles, say, when v is high enough for this term to be measurable then you ought to be using the relativistic formula for energy. Below that velocity the Newtonian approximation is good enough.
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