How was this function approximated? I found this passage in my book which is not very clear to me as to how it was obtained. 
Given $n\in\mathbb{N},\,k\in(0,1)$, then
$$(1+i)^{n+k}\approx(1-k)(1+i)^n+k(1+i)^{n+1}=(1+i)^n(1+ki)$$

Is $i$ supposed to be small? If so, the derivative of $(1+i)^{n}$ with respect to $n$ is $(1+i)^n\log(1+i)$. Thus a first order approximation of $(1+i)^{n+k}$ is 
$$
(1+i)^{n}+k(1+i)^{n}\ln(1+i). 
$$
If $0<i<1$ then $i\approx\ln(1+i)$ (again using a Taylor series approximation), and hence one arrives at the desired approximation of $(1+i)^{n}+ik(1+i)^{n}$.
