Help to understand theory of relativity?

I'm reading the book to undersatand how the thoery was made and so far reading very slowly I've understood up to the point where it speaks of kinetic energy, and it think it's the most important part. It reads "In accordance with the theory of relativity the kinetic energy of a material point of mass m is no longer the well known expression m*v^2/2'. But by the expression mc^2/√(1*v^2/c^2)." Then lastly it reads "If we develop the expression for the kinetic energy in the form of a sweries, we obtain mc^2 + mv^2/2 - 3/8mv^4/c^2+..." Can someone explain this to me please