What does the third term signify in series development?

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Vishwas K S
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How to develop in series as in the image below? What does the third term signify?
Printed page from The Meaning of Relativity, page 50, showing text and equation (45): E = m + (m/2)q^2 + (3/8)mq^4 + ..., the series expansion of energy in powers of q^2.
 
It is a simple Taylor expansion of the relativistic energy.
 
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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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