The Relativistic Kinetic Energy Formula

The energy of a particle relative to a rigid body is m_oc² / √1 - v² / c² where m_o is the mass relative to the particle, v is the velocity of the particle relative to the rigid body, and, c is the speed of light in a vacuum.  This can be expressed as an infinite series:

m_oc² + m_ov² / 2 + 3m_ov⁴ / (8c²) + ... 

This series minus the first term is the energy associated with motion.  It is equal to (γ - 1)m_oc² where γ = 1 / √1 - v² /  c².