Here are two slides from Fundamentals of Power Electronics. When talking about approximation roots on n-th order polynomial $P(s)=1+a_1s+...+a_ns^n$ with the form of $P(s)=(1+\tau_1s)(1+\tau_2s)...(1+\tau_3s)$, it uses some math that I cannot figure out on myself.
Firstly, I'd like to show the result when every root separate with each other.
And here is the result if consider two close roots exists.
In the last step, I do not know why approximation is accurate leads to the last inequality. Could you give me some hint or explanation? Thanks.
Here is the original slide file.