Find the Taylor series $z = 0$

Question

Problem is to find the taylor series in $z=0$ $$f(z) = {z \over z^2 + i}$$ What i tried $${d(ln(z^2 + i)) \over 2dz } = {z \over z^2 + i}$$ So I tried to find Taylor series of $ln(z^2 + i) \over 2$ and then differentiate it, but it doesn't seems right.

Any help or advice are welcome!

Answer 1

HINT:

$$\frac1{z+a}=\frac{1}{a}\sum_{n=0}^\infty (-1)^n\left(\frac za\right)^n$$