Note that the power series of cos x {\displaystyle \cos x} centred at x = 0 {\displaystyle x=0} is
converging for all real x . {\displaystyle x.}
Replacing x {\displaystyle x} by x 2 , {\displaystyle x^{2},} we obtain the power series of cos ( x 2 ) {\displaystyle \cos(x^{2})} centred at x = 0 , {\displaystyle x=0,} which converges for all real x {\displaystyle x} :