Question

f(z)=z/(z＋1)(z＋2)在z0=2处展开成泰勒级数,要详细步骤

Answer 1

先裂项

f(z)=z/(z＋1)(z＋2)=-1/(1+z)+2/(2+z)

再根据需要变项

f(z)=-1/(3+z-2)+2/(4+z-2)=(-1/3){1/[1-[(-1)(z-2)/3]}+(1/2){1/[1-[(-1)(z-2)/4]}

再展开,z-2的绝对值小于3(因为-1是奇点)

f(z)=sigma{(-1/3)[(-1)(z-2)/3]^n}+(1/2)[(-1)(z-2)/4]^n}

sigma是求和号,从n=0到无穷