将函数f(x)=x/(x-3)展开成x的幂级数．
f(x)=x/(3-x)，f(0)=0 f′(x)=3/(3-x)2 =3(3-x)-2，f′(0)=3×3-2 f′′(x)=3×2(3-x)-3，f′′(0)=3×2×3-3 f′′(x)=3×3×2(3-x)-4，f′′′(0)=3×3×2×3-4f(4)(x)=3×4×3×2(3-x)-5，f(4)(0)=3×4×3×2×3-5 ⋮ ⋮ f(n)(x)=3n! (3-x)-n-1 f(n)=3n!•3-n-1=n!•3-n． limn→∞Rn(x) =limn→∞[f(n+1)(ξ)/(n+1)!]xn+1 =limn→∞[(n+1)!•3(3-ξ)-n-2/(n+1)!]xn+1 =0(ξ,在0与x之间)． 收敛半径R=limn→∞∣an+1/an∣ =limn→∞∣n!•3-n/[(n+1)!•3-n-1]∣ =limn→∞3/(n+1)=0. f(x)=f(0)+f′(0)x+[f′′(0)/2!]x2+[f′′′(0)/3!]x3!+…+[f(n)(0)/n!]•xn+… =3-1x+3-2x2+3-3x3+…+3-nxn+…(收敛半径R=0)．
