Functions of Complex Variables - Find the first 4 terms of the Taylor Series.

Question

I have been asked the following question: Find the first four terms of the Taylor Series of the following function about 0. $$ f(z)=\frac{e^z}{(1+z)} $$ I know that the solution to this question is: $$ f(z) = 1 + 0z + (z^2)/2 - (z^3)/3 + ... $$ I have differentiated the original function and found values for each of the derivatives but all of my answers are positive so I have been unable to show that the fourth term is negative. Are there any other ways to solve this question other than just by repeated differentiation?

Answer 1

Separate...and you shall win. For $\;z\;$ close enough to zero:

$$e^z\cdot\frac1{1+z}=\left(1+z+\frac{z^2}2+\frac{z^3}6+\ldots\right)(1-z+z^2-z^3+\ldots)=$$

$$1+\frac12z^2-\frac13z^3+\frac76z^4+\ldots$$