f(x)=sin²x,先求它在x=0处的各阶导数。因为f(ⁿ)(x)=2^(n-1)sin[2x+(n-1)π/2]。所以
f(0)=0,f′(0)=0,f″(0)=2,f(³)(0)=0,f(4)(0)=-8,……,f(ⁿ)(0)=2^(n-1)sin[(n-1)π/2]
所以展开后为f(x)=2x²/2!-2³x^4/4!+2^5x^5/5!-2^7x^7/7!+……
+2^(n-1)sin[(n-1)π/2]xⁿ/n!+……