The kinetic energy of motion of a particle is the relativistic total energy minus the rest energy.

(a) A particle has rest mass $M$ and speed $v$. If $v \ll c$, then show that the kinetic energy due to motion is approximated by the well-known non-relativistic expression for the kinetic energy.

(b) Derive the leading correction to the non-relativistic expression.

In the above problem, I've done part (a) by expanding the Lorentz factor as a power series and then truncating the "negligible" terms.

However, I'm stumped for deriving the "leading correcton". I've read the Wikipedia article on leading corrections: https://en.wikipedia.org/wiki/Leading-order_term, but I still don't understand what the problem is asking for exactly?