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It's a taylor expansion of the gamma factor and the full relativistic energy/mass equivalence.
Second answer here explains better than I can: https://www.reddit.com/r/askscience/comments/2r0dtk/are_the_emc2_and_ek12mv2_equations_connected/
The later terms are negligible as they start to include division by increasing powers of c.
And physics stackexchange is fairly authoratative: https://physics.stackexchange.com/questions/435951/what-do-the-small-terms-in-the-series-expansion-of-relativistic-energy-mean
Backstop
mespilus
mv = momentum
1/2mv2 = kinetic energy
1/6mv3 = ?