Problem with Taylor Series

Question

So, I'm studying a book and got stuck in the following lines:

What should I do/use to get from the first line to the second one? How can I perform that kind of approximation?

I was thinking in this way: If I treat everything within the square brackets as $f(x)$, and I know its Taylor series, is there a way to get the Taylor series of $\frac{1}{f(x)}$?

Answer 1

Hint:

Use $ [{1-x}]^{-1}=1+x+ {x^2}+\dots$

with $x=(t_0-t)H_0+\frac12(t_0-t)^2q_0H_0^2+\dots$