If f is any function, analytic at x_{0} then
f(x)= f(x_{0})+ f '(x_{0})(x- x_{0})+ (f''(x_{0})/2)(x- x_{0})^{2}+ (f'''(x_{0})/6)(x- x_{0})^{3}+ ...
the general term is (f^{(n)}(x_{0})/n!)(x-x_{0})^{n} where f^{(n)} is the nth derivative.

In particular, use the "McLaurin series" which is the Taylor's series with x_{0}= 0 and f(x)= (1+ x^{2}/c^{2})^{-1/2}, set x= v, and then multiply by mc^{2}.